 Chinese Medicine, 2011, 2, 158-170 doi:10.4236/cm.2011.24026 Published Online December 2011 (http://www.SciRP.org/journal/cm) Copyright © 2011 SciRes. CM Mathematical Reasoning of Treatment Principle Based on “Yin Yang Wu Xing” Theory in Tr aditional Chinese Medicine (II) Yingshan Zhang School of Finance and Statistics, East China Normal University, Shanghai, China E-mail: ysh_zhang@163.com Received April 3, 2011; revised April 29, 2011; accepted May 15, 2011 Abstract By using mathematical reasoning, this paper demonstrates the treatment principle: “Do not treat a disease after it has occurred. But treat the disease before it will occur” (不治已病治未病) based on “Yin Yang Wu Xing” Theory in Traditional Chinese Medicine (TCM). We defined generalized relations and reasoning, in- troduced the concept of generalized steady multilateral systems, and discussed its energy properties. Later based on the treatment of TCM and treated the healthy body as a steady multilateral system, it has been proved that the treatment principle above is true. The kernel of this paper is the existence and reasoning of the non-compatibility relations in steady multilateral systems, and it accords with the oriental thinking model. Keywords: Traditional Chinese Medicine (TCM), “Yin Yang Wu Xing” Theory, Steady Multilateral Systems, Opposite Relations, Side-Effects, Medical and Drug Resistance Problem 1. Main Differences between Traditional Chinese Medicine and Western Medicine Western medicine treats disease at Microscopic point of view. It is will known that Western medicine always ma- kes directly medical treatments on disease organs after the disease of organs has occurred. The method always destroys the original human being’s balance and has none beneficial to human’s immunity. Western medicine can produce pollution to human’s body, having strong side-effects. Excessively using medicine can easily para- lysis the human’s immunity, which AIDS is a product of Western medicine. Using medicine too little can easily produce the medical and drug resistance problem. Traditional Chinese Medicine (TCM) studies the wo- rld from the Macroscopic point of view, and its target is in order to maintain the original balance of human being and in order to enhance the immunity. TCM has over 5000-year history. It almost has none side-effects or medical and drug resistance problem. TCM believes that a sick organ will lead other non-disease organ into illness if the sick organ continues to develop. The basic prince- ple of treatment is to cure the organs before the disease will occur in the organs rather than after. In other words, the basic principle of treatment is “Do not treat a disease after it has occurred. But treat the disease before it will occur” (不治已病治未病). After long period of practicing, Chinese ancient medi- cal scientists use “Yin Yang Wu Xing” Theory extensi- vely in the traditional treatment to explain the origin of life, human body, pathological changes, clinical diagno- sis and prevention. It has become an important part of TCM. “Yin Yang Wu Xing” Theory has a strong influ- ence to the formation and development of Chinese medi- cine theory. But, many Chinese and foreign scholars still have some questions on the reasoning of TCM. Zhang’s theories, multilateral matrix theory [1] and multilateral system theory [2-15], have given a new and strong mathematical reasoning method from macro (Glo- bal) analysis to micro (Local) analysis. He and his col- leagues have made some mathematical models and methods of reasoning [1-28], which make the mathema- tical reasoning of TCM possible based on “Yin Yang Wu Xing” Theory [29]. This paper will use steady multilate- ral systems to demonstrate the treatment principle of TCM above. The article proceeds as follows. Section 2 contains ba- sic concepts and main theorems of steady multilateral  159 Y. S. ZHANG systems while the treatment principle of TCM is demon- strated in Section 3. Some discussions in TCM are given in Section 4 and conclusions are drawn in Section 5. 2. Basic Concept of Steady Multilateral Systems In the real world, we are enlightened from some concepts and phenomena such as “biosphere”, “food chain”, “eco- logical balance” etc. With research and practice, by using the theory of multilateral matrices [1], and analyzing the conditions of symmetry [16,17] and orthogonality [18-28] what a stable system must satisfy, in particular, with analyzing the basic conditions what a stable working procedure of good product quality must satisfy [6,15,22], we are inspired and find some rules and methods, then present the logic model of analyzing stability of complex systems–steady multilateral systems [2-15]. There are a number of essential reasoning methods based on the sta- ble logic analysis model, such as “transition reasoning”, “atavism reasoning”, “genetic reasoning” etc. We start and still use the concepts and notations in papers [4,5]. 2.1. Generalized Relations and Reasoning Let V be a non-empty set and define its direct product as ,: ,VVxy xVyV VV .A non-empty subset is called a relation of V. Traditional Chinese Sci- ence (TCS) mainly researches general relation rules for general V rather than for special V. R For a relation class 0 ,, m RR i R ij R xR 1 , define both an inverse relationof and a relation multiplica- tion between and as follows: 1 i R ,:y ij RR 1 Rx R ,y ii ,: ij RR xy there exists at least an uV such that ,i uR and , uy R. The relationiis called reasonable ifR1 Ri . A gen- eralized reasoning of V is defined as forijthere is a relationsuch that RR . k R k Rij RR 2.2. Equivalence Relations Let V be a non-empty set with a relation R. The relation R is called an equivalence relation, denoted by ~, if the following three conditions are all true: (1) Reflexive: , xR for all V, i.e., x; (2) Symmetric: if , yR, then , xR, i.e., if y, then ; yx (3) Conveyable (Transitivity): if ,,, yRyzR, then , zR, i.e., if ,, yyz. then z Furthermore, the relation R is called a compatibility relation if there exists a non-empty subset 1 such that R1 satisfies at least one of the conditions above. And the relation R is called a non-compatibility relation if there doesn’t exist any non-empty subset 1 such that 1 satisfies any one of the conditions above. Any one of compatibility relations can be expanded into an equivalence relation [2]. RR RR R Western science only considers the reasoning under one Axiom system such that only compatibility relation reasoning is researched. However there are many Axiom systems in Nature. TCS mainly researches the general- ized reasoning among many Axiom systems in Nature. Of course, she also considers the reasoning under one Axiom system but she only expands the reasoning as the equivalence relation reasoning. 2.3. Two Kinds of Opposite Non-Compatibility Relations Equivalence relations, even compatibility relations, can not portray the structure of the complex systems clearly. In the following, we consider two non-compatibility relations. For example, let 04 VV V be a non-empty, th- ere are two kinds of opposite relations: the neighboring relation 1, denotedand the alternate relation, de- noted , having the property: R2 R (1) If ,, yy zthen , zi.e., if , y 1,R 1 , zR , then ,2 zR , or, ; 11 RR 2 R if ,, yx zthen i.e., if ,yz ,1 yR , 2 , zR , then ,1 zR , or, ; 1 12 RR1 R if ,, zy zthen , yi.e., if ,2 zR , 1 , zR , then ,1 yR or, . 1 21 RR 1 R (2) If ,, yy zthen i.e., if ,zx 2 ,, yR 2 ,, zR then ,zx 1 R , or, 1 1 R 22 RR ; if ,z x y, then,i.e., if yz ,zx 1 R , 2 , yR , then ,2 zR, or, 1 2 R 12 RR ; if , then ,yzzx y,i.e., if ,2 zR, 1 ,zx R , then ,2 yR , or, 1 R 21 2 RR . Two kinds of opposite relations can not be existence separately. Such reasoning can be expressed in Figure 1. The first triangle reasoning is known as a jumping-transi- tion reasoning, while the second triangle reasoning is known as an atavism reasoning. Reasoning method is a triangle on both sides decided to any third side. Both neigh- boring relations and alternate relations are non-compatibility relations (of course, non-equivalence relations), which are called two kinds of opposite non-compatibility relations. Figure 1. Triangle reasoning. Copyright © 2011 SciRes. CM  Y. S. ZHANG 160 2.4. Genetic Reasoning Let V be a non-empty set with an equivalence relation, a neighboring relation1and some alternate relations. Then a genetic reasoning is defined as follows: 0 R 2 RR (1) if , yy z,then z, i.e., if 0 , yR , ,1 zR ,, then 1 z , R, or, ; 01 RR 1 R (2) if yy z, then z, i.e., if ,, 1 yR ,0 zR ,, then 1 zR, or, ; 10 RR 1 R (3) if , yy z, then z, i.e., if ,, 0 yR ,2 zR ,, then 2 zR, or, ; 02 RR 2 R (4) if , yy z, then, z i.e., if , 2, yR ,0 zR ,, then 2 zR, or, . 20 RR 2 R 2.5. Multilateral Systems For a non-empty set V and its some relations, 01 ,,, :, m RR VVxyxVyV , Denote 0. Then the form (or sim- ply, V)is called a multilateral system [3,13], if ,, 1m RR ,V ,V satisfies the following properties: a) . 01m RRV ,RR RRR V b) . 00ii c) For any , we have i i ,,RR 0 R 1im 1 i R . d) For any , there existssuch that . kl RR i R kl RR R i The d) is called the generalized reasoning, the a) the uniqueness of reasoning, the b) the hereditary of reason- ing (or genetic reasoning) and the c) the condition of reasoning, respectively. In this case, the two-relation set is a lateral relation of V. Furthermore, the V and are called a state space and relation classes of , respectively. The generalized system 1 , ii RR ,V ,V can be written as 010 , ,, nm VVRR ,V 1 For a multilateral system , it is called complete (or, perfect) if “” changes into “=”.And it is called complex if there exists at least a non-compatibility rela- tion i. In this case, the multilateral system . R ,V is also called a logic analysis model of complex systems [2-15]. For a multilateral system ,V, assume that there exist relations: an equivalence relation, a neighboring relation, and an alternate relation in system V, which satisfy genetic reasoning. Suppose that for every, yV , at least there is one of the three relations between x and y, and there are not y and y R at the same time, i.e., 012 2112 VV 11 R ,RRR RR . Then it is easily to prove that V is a multilateral system with two non-compatibility relations, i.e., ,VR 01 4, where two non-compatibility rela- tions are the lateral relations and satisfying ,,,R 1 223 R ,R ,RR d VR ,RR 1 11 ,5 , ij 4 R ,ij mo R ij RR . The multilateral system ,V is equivalent to the logic architecture of reasoning model of “Yin Yang” Theory in Ancient China. In this paper, we only consider this multilateral system. Theorem 2.1. For a multilateral system ,V with two non-compatibility relations, , yV, only one of the following five relations is existent and correct: , y , , , yx yxyxy RR i.e., + 01 R RVV 2 R 34 . Theorem 2.2. For a multilateral system ,V with two non-compatibility relations, ,, yzV, the fol- lowing reasoning holds. (1) If , zy z, then y, i.e., ; 1 11 RR R 0 (2) if , zy z, then y,i.e., ; 1 RR R 22 0 (3) if , yx z, then ,i.e., ; yz1 RRR 11 0 (4) if , yx z, then ,i.e., . yz1 RRR 22 0 2.6. Steady Multilateral Systems The multilateral system V with two non-compatibility relations is said steady (or, stable) if there exists at least the chain 1,, n xV , which satisfies any one of the two conditions below: 12 , n1 xx 10 n RR xi.e., there exists a number n such that where ; 11 n n RR R 1 112 , n xx 20 n RR x i.e., there exists a number n such that where . 22 n n RR R 2 Theorem 2.3. For a steady multilateral system ,V = 010 ,, nm VVRR 5 10 RR 1 with two non-compati- bility relations, there exists five-length chain, and the length of the chain is integer times of 5, i.e., there exists 5 20 RR (or ) and 5kn if 10 n RR 20 n RR (or ). Theorem 2.4. For a steady multilateral system ,V = 010 ,, nm VVRR 1 with two non-compati- bility relations, assume there exists a chain 012 ,,, xx 34 , x5nm, then . In other words, 5m and there exists a partition of V as follows: 01234 VVVVVV , ,0,, ii VyVyxi 4. 2 Theorem 2.5. For the decomposition above for the steady multilateral system with two non-compatibility relations, there exist relations below Figure 2. In other words, 55 10 RRR , i.e., is a complete mul- tilateral system. ,V Theorem 2.6. For each element V in a steady multilateral system V with two non-compatibility rela- tions, there exist five equivalence classes below: Copyright © 2011 SciRes. CM  161 Y. S. ZHANG | yVy x , | S yVx y , | K yVx y , | X yVy x |Vyx , X Sy which the five equivalence classes have relations below Figure 3. These theorems can been found in [2-15]. Figures 2 and 3 in Theorems 2.5 and 2.6 are the Figures of “Wu Xing” Theory in Ancient China. The steady multilateral system V with two non-compatibility relations is equiva- lent to the logic architecture of reasoning model of “Yin Yang Wu Xing” Theory in Ancient China. What de- scribes the general method of complex systems can be used in human complex systems 3. Relationship Analysis of Steady Multilateral Systems 3.1. Energy Theory of Multilateral Systems Western Medicine is different from TCM because the TCM has a concept of Chi （or Qi, 气） as a form of energy. It is believed that this energy exists in all things (living and non-living) including air, water, food and sunlight. Chi is said to be the unseen vital force that Figure 2. Uniquely steady architecture of Wu-Xing. Figure 3. The method of finding Wu-Xing. nourishes one’s body and sustains one’s life. It is also believed that an individual is born with an original amount of Chi at the beginning of one’s life and as one grows and lives, one acquires Chi from eating and drinking, from breathing the surrounding air and also from living in one’s environment. And the one also af- fords Chi for the human body’s meridian system (Jing-Luo (经络)) and Zang Xiang（藏象） organs which form a parasitic system of human, called the second physiological system of human. The second system of human controls the first physiological system (Anatomy system) of human. An individual would become ill or dies if one’s Chi in the body is imbalanced or exhausted. The concept is summarized as the energy theory of mul- tilateral systems. In mathematics, a multilateral system is said to have Energy (or Dynamic) if there is a non-negative function which makes every subsystem meaningful of the multilateral system. Similarly to papers [4,5], unless sta- ted otherwise, any equivalence relation is the liking rela- tion, any neighboring relation is the loving relation, and any alternate relation is the killing relation. Suppose V is a steady multilateral system having en- ergy during normal operation. Then its energy function for any subsystem of the multilateral system has a center (or average, expected value, median in Statistics). And this state is called normal when the energy function is nearly to the center. Normal state is the better state. A subsystem of a multilateral system is called not run- ning properly (or disease, abnormal) if the energy devia- tion from the center of the subsystems is too large, the high (real disease) or the low (virtual disease). In general, a disease is less serious if it satisfies both the loving relation and the killing relation of the multi- lateral system. This disease is less serious because this disease has not undermine the balance of the multilateral system and the normal order, which makes the interven- tion did not reduce the intervention reaction coefficient 1 (See [4,5]). But the disease is serious if it doesn’t satisfy both the loving relation and the killing relation of the multilateral system, i.e., there is an incest order. This disease is serious because the disease has destroyed the balance and the normal order of the multilateral system, which makes the intervention reaction coefficient 1 reduced response to intervention. For a steady multilat- eral system V with two non-compatibility relations, sup- pose that the subsystems X,,,, SKXX XKSare the same as those defined in Theorem 2.6. Then the diseases can be decomposed into the following classes: Definition 3.1. (involving(相及) and infringing upon (相犯)) Suppose a disease occurred between X and S . The disease is called less serious if X is virtual disease and so is S at the same time (Mother disease involv- Copyright © 2011 SciRes. CM  Y. S. ZHANG 162 ing the son母病及子), or, if X is real disease and so is S at the same time (Son disease infringing upon the mother 子病犯母). The disease is called rare disease if X is real disease but S is virtual disease at the same time, or if X is virtual disease and S is real disease at the same time. Definition 3.2. (multiplying(相乘) and insulting(相侮)) Suppose a disease occurred between X and . The disease is called less serious if X is real disease and is virtual disease at the same time. The rela- tion between X and is called multiplying-relation (相乘). The disease is called serious if X is virtual disease but is real disease at the same time. The relation be- tween X and is called insulting-relation (相侮). It means that has harmed the X by using the method of incest. The disease is called rare disease if X is real disease and so is at the same time, or if X is virtual disease and so is at the same time. The disease is called more serious if X is real disease, and is virtual disease but X is also virtual dis- ease at same time, i.e., X not only multiplies in , but also X insults X by using the method of incest. It is because the energy of X is too high. The relation between X and is also called multiplying-insulting relation (相乘致侮). The disease of multiplying-insulting relation will re- sult in more than three subsystems falling-ill. Generally, three or more subsystems falling-ill, it will be hard to cure. Therefore, the multiplying-insulting disease should be avoided as much as possible. In Chinese words, it is that “Again and again, not only to the repeated to four “(只有再一再二，没有再三再四)―Allow one or two subsystems fall ill, but don’t allow three or four subsys- tems fall ill. Theorem 3.1. The occurrence of disease has its laws: The first occurrence of the loving relation and the killing relation after the disease. In other words, the following statements are true. If a subsystem X of a multilateral system V falls virtual disease, the law is the first occurrence of the less serious virtual disease of the mother X, and the less serious real disease of the bane X S after the disease, and next the more serious real disease of the prisoner , and finally the less serious virtual disease of the son S . If a subsystem X of a multilateral system V falls real disease, the law is the first occurrence of the less serious real disease of the son S , and the less serious virtual disease of the prisoner after the disease, and next the more serious virtual disease of the bane X , and finally the less serious real disease of the mother . X It is because, by Theorem 3.3 below, the energy of the son S S or the mother X of a falling-ill subsystem X of a multilateral system V is firstly changed by following the energy change of X corresponding to the real or vir- tual disease, respectively. If the disease continues to de- velop, it will lead to undermine the capability of self-protection of the multilateral system, i.e., the self- protection coefficient 3 S will near to small. By Theo- rem 3.2 below, the victim or X of X will en- counter the sick which is different from the disease di- rection of X. In a subsystem of a multilateral system being not run- ning properly, if this sub-system energy increases or de- creases through external force, making the energy return to the center (or average, expected value, median), this method is called intervention(or making a medical treat- ment) to the multilateral system. The purpose of intervention is to make the multilateral system return to normal state. The method of intervene- tion is to increase or decrease the energy of a subsystem. What kind of treatment should follow the principle to treat it? For example, Western medicine always makes directly medical treatments on disease organs after the disease of organs has occurred. But TCM always makes indirectly medical treatments on disease organs before the disease of organs will occur. In mathematics, which is more reasonable? Based on this idea, many issues are worth further dis- cussion. For example, if an intervention treatment has been done to a non-sick subsystem of a multilateral sys- tem, what situation will happen? 3.2. Intervention Rule of a Multilateral System For a steady multilateral system V with two non-compati- bility relations, suppose that there is an external force (or an intervening force) on the subsystem X of V which makes the energy () changed by increment () () , then the energies X (),( ),(), SKX XKS ,,, SKXX of other subsystems XKS (defined in Theorem 2.6) of V will be changed by the increments () S , , K X , X K X S , respectively. The concepts of the capability of intervention reaction, beneficiaries and victims come from papers [4,5]. In general, the intervention rule is similar to force and reac- tion in Physics. In other words, if a subsystem of multi- lateral system V has been intervened, then the energy of subsystem which has neighboring relation ( or benefici- ary ) changes in the same direction of the force, and the energy of subsystem which has alternate relation (or vic- tim ) changes in the opposite direction of the force. The size of the energy changed is equal, but the direction op- posite. In mathematics, the changing laws of intervention rule Copyright © 2011 SciRes. CM  163 Y. S. ZHANG are as follows. (1) If , then 0X 1S X , K =1 X K, 2 2X S, X ; (2) If , then 0X 2S , X2K X K, 1 1X S, ; where 12 10 . Both1 and 2 are called inter- vention reaction coefficients, which are used to represent the capability of intervention reaction. The larger1 , the better the capability of intervention reaction. The state 11 is the best state but the state 10 is the worst state. Medical and drug resistance problem is that such a question, beginning more appropriate medical treatment, but is no longer valid after a period. It is because the ca- pability of intervention reaction is bad, i.e., the interven- tion reaction coefficients 1 is too small. In the state 11 , any medical and drug resistance problem is non- existence but in the state10 , medical and drug resis- tance problem is always existence. At this point, the pa- per advocates the principle of treatment to avoid medical and drug resistance problems. 3.3. Self-Protection Rule of a Multilateral System If there is an intervening force on the subsystem X of a steady multilateral system V which makes the energy () be changed by increment , XX such that the energies ,, SK XK , S S of other sub- systems ,X ,X (defined in Theorem 2.6) of V will be changed by the increments S S , , , respectively, then can the multilateral system V have capability to protect the worst victim to restore? , X K X S The concepts of the general capability of self-protect- tion, the better capability of self-protection and worst victims come from papers [4,5]. In general, there is an essential principle of self-protection: any harmful sub- system of X should be protected by using the same inter- vention force but any beneficial subsystem of X should not if the intervention treatment has been done on the subsystem X of the multilateral system. In order to represent the capability of self-protection, the concept of the self-protection coefficient3 is also need. The larger3 , the better the capability of self-pro- tection. The state 31 is the best state but the state 31 0 is the worst state, where 1 is the interven- tion reaction coefficient. In the case of virtual disease, the treatment method of intervention is to increase the energy. If the treatment has been done on X by the increment , the worst victim is the prisoner 0X of X which has the increment 1 . Thus, the treatment of self-protection is to restore the prisoner of X and the restoring method of self-protection is to increase the energy of the prisoner of X by using the interven- tion force on X according to the intervention rule. But there may be an increase degree which is 3 rather than 1 if the capability of self-protection 3 is less than 1 . In the case of real disease, the treatment method of in- tervention is to decrease the energy. If the treatment has been done on X by the increment , the worst victim is the bane X 0X of X which has the incre- ment 1 . Thus, the treatment of self-protection is to restore the bane X of X and the restoring method of self-protection is to decrease the energy X of the bane X of X by using the same intervention force on X according to the intervention rule. But there may be an decrease degree which is 3 1 rather than if the capability of self-protection 3 is less than 1 . In mathematics, the following self-protection laws hold. (1) If X 0 , then the energy of subsystem will decrease the increment 1 ()0 , which is the worst victim. So the capability of self-protection in- creases the energy of subsystem by increment 3 in order to restore the worst victim by ac- cording to the intervention rule. (2) If X 0 , then the energy of subsystem X will increase the increment 1, which is the worst victim. So the capability of self-protection de- creases the energy of subsystem X by increment 3 in order to restore the worst victim X by according to the intervention rule. In general, for an usual body, by logic and practice, we found that 231 and that 1 nears to 512 0.618 and that 2 nears to 2 1151 2 0.382 and that 3 nears to 12 121 2 0.5 . Interestingly, they near to the golden numbers. Theorem 3.2. Suppose that a steady multilateral sys- tem V which has energy and capability of self-protection is with both intervention reaction coefficients and self- protection coefficient 1 ,2 and 3 . If the capability of self-protection wants to restore both subsystems and X , then the following statements are true. (1) In the case of virtual disease, the treatment method is to increase the energy. If an intervention force on the subsystem X of steady multilateral system V is imple- mented such that its energy has been changed by increment X 0 , then all five subsystems will be changed finally by the increments as follows: 23 21 10XXX , 123 21 0, SSS XXX 13 21 , KKK XXX Copyright © 2011 SciRes. CM  Y. S. ZHANG 164 213 21 , XXX KKK 213 21 , XXX SSS 0X (1) (2) In the case of real disease, the treatment method is to decrease the energy. If an intervention force on the subsystem X of steady multilateral system V is imple- mented such that its energy ' 0 has been changed by increment , then all five subsys- tems will be changed finally by the increments as fol- lows: 'X 23 21 '' '1XXX 0, 0 , 0 213 21 '' ' SsS XXX , 213 21 '' ', kkk XXX 13 21 '' ', XXX KKK 123 21 '' ' XXX SSS 'X (2) Corollary 3.1. Suppose that a steady multilateral sys- tem V which has energy and capability of self-protection is with both intervention reaction coefficients and self- protection coefficient 1 ,2 and 3 . Then the capa- bility of self-protection can make both subsystems and X to be restored at the same time, i.e., the capability of self-protection is better, if and only if 2 21 and 31 . Side effects of medical problem were the question: in the medical process, destroyed the normal balance of non-fall ill system or non-intervention system. By Theo- rem 3.2 and Corollary 3.1, it can be seen that if the capa- bility of self-protection of the steady multilateral system is better, i.e., the multilateral system has capability to protect all the victims to restore, then a necessary and sufficient condition is 2 21 and 31 . At this point, the paper advocates the principle to avoid any side-effects of treatment. 3.4. Mathematical Reasoning of Treatment Principle by the Neighboring Relations of Steady Multilateral Systems In order to show the rationality of the treatment principle above, it is needed to prove the following theorems. Theorem 3.3. Suppose that a steady multilateral sys- tem V which has energy and capability of self-protect- tion is with both intervention reaction coefficients and self-protection coefficient 1 , 2 and 3 satisfying 21 In the case of virtual disease, if an intervention force on the subsystem X of steady multilateral system V is implemented such that its energy increases the increment 0X , then the subsystems X, S and X can be restored at the same time, but the subsystems X and S will increase their energies by the finally increments 23 21 X 3 1 1 , and 2 S X X 12 3 3 11 , respectively. On the other hand, in the case of real disease, if an intervention force on the subsystem X of steady multilat- eral system V is implemented such that its energy ' decreases, i.e., by the increment 'X 0 , the subsystems S , and X can also be restored at the same time, and the subsystems X and X will decrease their energies, i.e., by the finally incre- ments S 2 '1' 23 X 12 3 2 ' S 3 1 X 3 1 1, 1 ', and X respec- tively. Theorem 3.4. For a steady multilateral system V which has energy and capability of self-protection, as- sume both intervention reaction coefficients and self- protection coefficient is are 1 ,2 and 3 which satisfy 2 21 , 31 and 10 where 0 (th e following the same) is the solution of 0.589754 3 5123 11 2 1 . Then the following statements are true. (1) If an intervention force on the subsystem X of steady multilateral system V is implemented such that its energy has been changed by increment X 0 , then the final increment 3 11 of the energy S of the subsystem S changed is greater than or equal to the final increment 3 1 1 of the energy of the subsystem X changed based on the capability of self-protection. (2) If an intervention force on the subsystem X of steady multilateral system V is implemented such that its energy has been changed by increment X 0, then the final increment 3 11 of the energy X S of the subsystem X cha- nged is less than or equal to the final increment S 3 1 1 of the energy of the subsystem X changed based on the capability of self-protection. Corollary 3.2. For a steady multilateral system V which has energy and capability of self-protection, both intervention reaction coefficients and self-protection coefficient are 1 , 2 and 3 which satisfy 2 21 , 31 and 10 . Then the following statements are true. (1) If an intervention force on the subsystem X of steady multilateral system V is implemented such that its energy has been changed by increment X 0 , then the final increment 3 11 of the energy S of the subsystem S changed is 3 and 31 . Then the following statements are true. Copyright © 2011 SciRes. CM  Y. S. ZHANG 165 less than the final increment of the energy 3 1 1 of the subsystem X changed based on the capa- bility of self-protection. (2) If an intervention force on the subsystem X of steady multilateral system V is implemented such that its energy has been changed by increment , then the final increment 0X 3 11 of the energy of the subsystem changed is greater than the final increment of the energy X S X S 3 1 1 of the subsystem X changed based on the capability of self-protection. 3.5. Mathematical Reasoning of Treatment Principle by the Alternate Relations of Steady Multilateral Systems In order to show the rationality of the treatment principle above, it is needed to prove the following theorem. Theorem 3.5. Suppose that a steady multilateral sys- tem V which has energy and capability of self-protection is with both intervention reaction coefficients and self-pro- tection coefficient 1 ,2 and 3 satisfying 2 21 and 31 . Then the following statements are true. Assume there are two subsystems X and of V with an alternate relation such that X encounters virtual disease, and at the same time, befalls real disease. If an intervention force on the subsystem X of steady multilateral system V is implemented such that its energy has been changed by increment and at the same time, another intervention force on the subsystem X0 of steady multilateral system V is also implemented such that its energy 0 has been changed by increment , then all other subsystems: and S K X , XX KS can be restored at the same time, and the subsystems X and will increase and decrease their energies by the same size but the direction opposite, i.e., by the finally increments and X 23 1 3 1 1 3 0, 3 K X = respectively. 1 3 1 23 1 Assume there are two subsystems X and X 0, of V with an alternate relation such that X encounters real disease, and at the same time, X befalls virtual dis- ease. If an intervention force on the subsystem X of steady multilateral system V is implemented such that its energy has been changed by increment , and at the same time, another inter- vention force on the subsystem X X 0 of steady multilateral system V is also implemented such that its energy X 0 has been changed by increment X, then all other subsystems: and S K k , X SX can be restored at the same time, and the subsystems X and X will decrease and increase their energies by the same size but the direction opposite, i.e., by the finally incre- ments 3 23 1 311X 0, and 3 X K = 23 1 3 1 10 , respectively. 4. Rationality of Treatment Principle of TCM 4.1. Treatment Principle if Only One Organ of the Human Body System Falls Ill In the paper [4], it is proved that the following statements are true. (1) If only one subsystem falls ill, mainly the treatment method should be to intervene it indirectly for case: the capability coefficient 10 of intervention reaction, according to the treatment principle of “Real disease is to rush down his son but virtual disease is to fill his mother”. (2) The intervention method directly can be used in case 10 but should be used as little as possible. We see: The (1) is such an intervention when the in- tervention system was non-disease, but the incidence of law by this system will be to disease if the incidence of the disease continues to develop. The method means the idea of “Do not treat the disease after it has occurred. But treat the disease before it will occur”. Above are the main treatments. But also need to add adjuvant therapies because it is difficult to judge 10 S S S . In the case of virtual disease of X, by Theorem 3.1, the virtual disease of the loving subsystem X will first occur. Therefore, the primary treatment is to increase the energy of X before the virtual disease of X will occur. The next more serious illness which will occur is real disease of because will insult X if the disease of X continues to develop. By Theorem 3.5, the second treatment method of “Strong inhibition of the same time, support the weak” should be done for X and as follows: (3) Increase the energy of X, and at the same time, de- crease the energy of as the adjuvant therapy of increasing the energy of . X The method of the adjuvant therapy also means the idea of “Do not treat the disease after it has occurred. But treat the disease before it will occur ” since now the S doesn’t fall ill yet but it will be incidence. In the case of real disease of X, by Theorem 3.1, the virtual disease of the loving subsystemS will first occur. Therefore, the primary treatment is to decrease the en- ergy of S before the real disease ofS will occur. The next more serious illness which will occur is virtual disease ofX because X will insultX if the disease of X continues to develop. By Theorem 3.5, the treatment method of “Strong inhibition of the same time, support the weak” should be done for X andX as follows: Copyright © 2011 SciRes. CM  Y. S. ZHANG 166 (4) Decreases the energy of X, and at same time, in- creases the energy of X as the adjuvant therapy of decreasing the energy of S . The method of the adjuvant therapy also means the idea of “Do not treat the disease after it has occurred. But treat the disease before it will occur” since now the X doesn’t fall ill yet but it will be incidence. Above methods of the adjuvant therapy always can be done for any intervention reaction coefficient1 . By Theorems 3.2 and 3.5, this methods of the adjuvant therapy will be better if intervention reaction coefficient 2310 because the increment 23 1 will be larger. The intervention method can be to maintain the bal- ance of human body system because only both disease organs and intervention organs are treated according to the normal Wu-Xing order of human body, by Theorems 3.3 and 3.5, such that there is not any side effect for all other organs. The treatment method also increases the capabilities of intervention reaction and self-protection because the method has used them to treat disease. The function of human tissue, the stronger the more you use. Therefore the method will maintain the balance of the five subsys- tems of human tissue and makes the 1 and 3 near to 1, at the same time, the 2 near to 2 1 . The state 13 1 is the best state of the human body system. On the way, it almost has none medical and drug resis- tance problem since any medicine is possible good for some large 1 ,2 and 3 . 4.2. Treatment Principle if Only Two Organs with the Loving Relation of the Human Body System Encounter Sick 1) Suppose that the two organs X and S of the human body system are abnormal (or disease). In the human body of relations between two non-compatible relations with the constraints, only two normal situations may oc- cur: (1) X encounters virtual disease, and at the same time, S befalls virtual disease, i.e., the energy of X is too low and so is the energy of S . (2) X encounters real disease, and at the same time, S befalls real disease, i.e., the energy of X is too high and so is the energy of S . In paper [4], it has be shown that if intervention reac- tion coefficients satisfy31 ,2 21 and10 , then the following statements are true. (1) If one wants to treat the abnormal organs X and S for virtual disease, then the one should intervene the organ X directly by increasing its energy. It means that “Virtual disease is to fill his mother”. (2) If one wants to treat the abnormal organs X and S for real disease, then the one should intervene or- gan S directly by decreasing its energy. It means that “Real disease is to rush down his son”. Above are the main treatments. But also need to add adjuvant therapies because it is difficult to judge that 31 ,2 21 and 10 . In the case of virtual disease, since the diseases of the loving subsystems S and X have occurred, by Theo- rem 3.1, the next more serious illness which will occur is real disease of X because X will insult S if the disease continues to develop. By Theorem 3.5, since the method of increasing the energy of X has been done and a hope is that this method also can increase the energy of S , the treatment method of “Strong inhibition of the same time, support the weak” should be done for S and X as follows: (3) Increase the energy of S , and at the same time, decrease the energy of X as the adjuvant therapy of increasing the energy of X. The method of the adjuvant therapy also means the idea of “Do not treat the disease after it has occurred. But treat the disease before it will occur” since now the X doesn’t fall ill yet but it will be incidence. In the case of real disease, since the disease of the loving subsystems X andS have occurred, by Theorem 3.1, the next more serious illness which will occur is virtual disease of X because X will insult X if the disease continues to develop. By Theorem 3.5, since de- crease the energy of S has been done and a hope is that this method can decrease the energy of X, the treat- ment method of “Strong inhibition of the same time, support the weak” should be done for X and X as follows: (4) Decrease the energy of X, and at the same time, increase the energy of X as the adjuvant therapy of decreasing the energy of S . The method of the adjuvant therapy also means the idea of “Do not treat a disease after it has occurred. But treat the disease before it will occur” since now the X doesn’t fall ill yet but it will be incidence. Above methods of the adjuvant therapy always can be done for any the capability of intervention reaction coef- ficient 1 . By Theorems 3.2 - 3.5 and Corollaries 3.1 - 3.2, this adjuvant therapy will be better if the interven- tion reaction coefficients satisfies2310 be- cause the increment 23 1 will be larger. The intervention method also can be to maintain the balance of human body system because only both disease organs and intervention organs are treated according to the normal Wu-Xing order of human body, by Theorems 3.2 - 3.5, such that there is not any side-effects for all other organs. Copyright © 2011 SciRes. CM  167 Y. S. ZHANG The treatment method also increase the capability of intervention reaction because the method of using the capability of intervention reaction and capability of self-protection will maintain the balance between the five subsystems of human tissue and makes the 1 and 3 near to 1, at the same time, the 2 near to 2 1 . The state 13 1 is the best state of the human body system. On the way, it is almost none medical and drug resistance problem since any medicine is possible good for some large 1 . 2) Suppose that the disease is rare disease, i.e., X is real disease but S is virtual disease at the same time, or X is virtual disease and S is real disease at the same time. In this case, the 231 should be small, such as 10 , because the order of both loving relation and killing relation of the system has been de- stroyed to some extent. The treatment method of “Real disease is to rush down itself” (实则泻之) or “virtual disease is to fill itself” (虚则补之) should be directly done for both X and S . For the first case, if the methods of both decreasing the the energy of X and increasing the energy of S have been done when 10 , they will be to lead that kills X and promotes the order of the killing relation of the multilateral system although the methods will produce a certain amount of side-effects: S X 0 S 123and 123K. If the above method uses the future, making the physical side-effects occur really, then it means that the ability of the intervention and self-protection capability has been restored, i.e., the 231 0 X come to larger. When 10 1 , both the main treatment and the adjuvant therapy can be used by using the following methods: (5) Firstly decrease the energy of X and increase the energy of S at the same time. (6) Secondly increase the energy of X and de- crease the energy of X at the same time. The method can continue to promote the 1 because the capabilities of both intervention reaction and self- protection have been used. For the second case, if the methods of both increasing the energy of X and decreasing the energy of S have been done when 10 , there doesn’t exist any side- effects although there are only the small increments 231 12 0,X S X 231 12 0 and . If the above method uses the future, making the more serious disease occur really, then it means that the capa- bilities of both the intervention reaction and self-protect- tion have been restored, i.e., 23 1 12 0, or the case: 2 21 , 31 and 10 . In this time, the adjuvant therapy can be used by using the following methods: (7) Increase the energy of X and decrease the en- ergy of S at the same time, as the adjuvant therapy of increasing the energy of X and decreasing the energy of S . The method can continue to promote the 1 because the capabilities of both intervention reaction and self-protection have been used. But the method should be used as little as possible because it may be to lead that X will insult S and destroys the killing order of the human body system if it always keen to be used. 4.3. Treatment Principle if Only Two Organs with the Killing Relation of the Human Body System Encounter Sick 1) Suppose that the organs X and of a human body system are abnormal (or disease). In the human body system of relations between two non-compatible rela- tions with the constraints, only a serious disease may occur: X encounters virtual disease, and at the same time, befalls real disease, i.e., the energy of X is too low and the energy of is too high. Such a disease has a serious problem because insults X. By Theorem 3.1, the disease most likely to lead to more serious illnesses: the virtual disease of and the real disease of X X S . In paper [4], it has been shown that if intervention re- action coefficients satisfy 1 2 2 , then one should intervene organ X directly by increasing its energy, and at the same time, intervene organ directly by de- creasing its energy if the one wants to treat the abnormal organs X and . It means that“Strong inhibition of the same time, support the weak”. Of course, a method of the adjuvant therapy always can be done as follows: (1) Increase the energy of X and decrease the en- ergy of X S at the same time. It means that “Virtual disease is to fill his mother but real disease is to rush down his son”. The method can continue to promote the1 because the capabilities of both intervention reaction and self- protection have been used. 2) If a less serious disease has occurred: X encounters real disease, and at the same time, befalls virtual disease, i.e., the energy of X is too high and the energy of is too low. Such a disease has not a serious problem in this time yet, but by Theorem 3.1 the disease most likely to lead to more serious illnesses: the virtual disease of X and the real disease of S . Therefore, this dis- ease should be done to treat as more serious diseases. The primary treatment method of “Strong inhibition of the same time, support the weak” should be done for X, and X , i.e., to decrease the energy of X, and at the same time, to increase the energies of and X . Copyright © 2011 SciRes. CM  Y. S. ZHANG 168 The method also means the idea of “Do not treat a dis- ease after it has occurred. But treat the disease before it will occur”. It is because the X is not sick at this time, but it will be incidence. And secondary treatment is as follows: (2) Decrease the energy of the subsystem S . It means that “Real disease is to rush down his son”. Above methods of the adjuvant therapy always can be done for any intervention reaction coefficient1 . By Theorems 3.2 - 3.5 and Corollaries 3.1 - 3.2, the methods of the adjuvant therapy will be better if the intervention reaction coefficients satisfies 0231 23 1 be- cause at this time the increment was lar- ger. The intervention method can be to maintain the bal- ance of human body system because only both disease organs and intervention organs are treated according to the normal Wu-Xing order of human body, by using Theorems 3.2 - 3.5, such that there is not any side-effects for all other organs. The treatment method also increase the capability of intervention reaction because the method of using both the capability of intervention reaction and capability of self-protection will maintain the balance between the five subsystems of human tissue and makes the 1 and 3 near to 1, at the same time, the 2 near to 2 1 . The state 13 1 is the best state of the human body system. On the way, it is almost none medical and drug resistance problem since any medicine is possible good for some large 1 . 3) Suppose that the disease is rare disease, i.e., X is real disease and so is at the same time, or if X is virtual disease and so is at the same time. In this case, the 231 should be small, such as 10 , because the order of both the loving relation and the killing relation of the system has been destroyed to some extent. The directly treating method of “Real disease is to rush down itself” (实则泻之) or “virtual disease is to fill itself” (虚则补之) should be directly done for X and . For the first case, if the methods of decreasing the en- ergies of both X and have been done when 10 , the methods will produce a certain amount of side- effects: and 0 S X 123 X S . If the above method uses the future, making the physical side-effects occur, then it means that the ability of the intervention and self-protection capabil- ity has been restored, i.e., the 231 123 0 comes to larger. When 10 , the adjuvant therapy can be used by using the following methods: (3) Decrease the energies of S and X at the same time. It means that “Virtual disease is to fill his mother but real disease is to rush down his son”. The method can continue to promote the 1 because the capabilities of both intervention reaction and self- protection have been used. For the second case, if the methods of increasing the en- ergies of both X and have been done when 10 , the methods will produce a certain amount of side- effects: 123 0 S X and X K 0 123 . If the above method uses the future, making the physical side-effects occur, then it means that the ability of the intervention and self-protection capabil- ity has been restored, i.e., the 231 come to larger. When 10 , the adjuvant therapy can be used by using the following methods: (4) Increase the energies of X and S S at the same time. It means that “Virtual disease is to fill his mother but real disease is to rush down his son”. The method can continue to promote the 1 because the capabilities of both intervention reaction and self- protection have been used. 5. Conclusions This work shows how to treat the diseases of a human body system and a lot of methods are presented. If only one organ falls ill, the treatment method should be to intervene it indirectly by using primary treatment and secondary treatment. (1) In the case of virtual disease of X, the primary treatment is to increase the energy of the loved subsys- tem X. And the secondary treatment is to increase the energy of X, and at the same time, to decrease the energy of S as the adjuvant therapy of increasing the energy of . X (2) In the case of real disease of X, the primary treat- ment is to decrease the energy of the loving subsystem S S . And the secondary treatment is to decrease the en- ergy of X, and at the same time, to increase the energy of X as the adjuvant therapy of decreasing the energy of S . If only two organs X and S of the human body system fall ill, the treatment method should be to inter- vene it indirectly by using primary treatment and secon- dary treatment. (1) Suppose that X encounters virtual disease, and at the same time, S befalls virtual disease. The primary treatment is to increase the energy of the subsystem X. And secondary treatment is to increase the energy of S , and at the same time, to decrease the energy of X as the adjuvant therapy of increasing the energy of X. (2) Suppose that X encounters real disease, and at the same time, S befalls real disease. The primary treat- ment is to decrease the energy of the subsystem S . Copyright © 2011 SciRes. CM  169 Y. S. ZHANG And secondary treatment is to decrease the energy of X, and at the same time, to increase the energy of X as the adjuvant therapy of increasing the energy of X. (3) Suppose that X encounters real disease, and at the same time, S befalls virtual disease (rare disease). The mainly treatment is to decrease the energy of the subsystem X, and at the same time, to increase the energy of the subsystem S if 10 . When 10 1 , the adjuvant therapy is firstly to decrease the energy of X and to increase the energy of S at the same time; and secondly to increase the energy of X and to decrease the energy of X at the same time. (4) Suppose that X encounters virtual disease, and at the same time, S befalls real disease (rare disease). The primary treatment is to increase the energy of the subsystem X, and at the same time, to decrease the en- ergy of the subsystem S if 10 . When 10 S , the secondary treatment is to increase the energy of X and to decrease the energy of at the same time, as the adjuvant therapy of increasing the energy of X and decreasing the energy of S . If only two organs X and of the human body system fall ill, the treatment method should be to inter- vene it indirectly by using primary treatment and secon- dary treatment. (1) Suppose that X encounters virtual disease, and at the same time, befalls real disease. The primary treatment is to increase the energy of the subsystem X, and at the same time, to decrease the energy of the sub- system . And secondary treatment is to increase the energy of the subsystem X and to decrease the sub- system S X at the same time. (2) Suppose that X encounters real disease, and at the same time, befalls virtual disease (may be to occur more serious disease). The primary treatment is to de- crease the energy of the subsystem X, and at the same time, to increase the energies of both and X . And secondary treatment is to decrease the energy of the subsystem S . (3) Suppose that X encounters real disease, and at the same time, be falls real disease (rare disease). The mainly treatment is to decrease the energies of the sub- systems X and if 10 . When 10 S , the adjuvant therapy is to decrease the energies of and X at the same time. (4) Suppose that X encounters virtual disease, and at the same time, befalls virtual disease (rare disease). The mainly treatment is to increase the energies of the subsystems X and if 10 . When 10 X S , the adjuvant therapy is to increase the energies of and S at the same time. Other properties, such as the order of “Wu Xing” (五 行), “Wu Yun Liu Qi” (五运六气), “Zang Xiang” (藏象), “Jing Luo” (经络), and so on, will be discussed in the next articles. 6. Acknowledgements This article has been repeatedly invited as reports, such as people’s University of China in medical meetings, Shanxi University, Xuchang College, and so on. 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