**Preliminaries** **Ⅲ****.1.** Before some characters of axiom1 are described,some algorithms are adopted to illustrate conveniently the relation among these quantities.The relation between two quantities are described by the term called ‘include’ and ‘equal divided’.

**Example** **Ⅲ****.2.** For example,Considering 10 and 2,That how much 10 includes 2 is expressed as 10（）2=5,and that how much 10 is equally divided by 2 is expressed as 10/2=5 . 10（）2=5 is not the same as 10/2=5 because the former can be expressed as 10/5=2 by ‘equal divided’ and the latter can be expressed as 10（）5=2 by ‘include’.

**Theorem ****Ⅲ****.3.** I find that the extension only can be executed in the way of unit superposition in the system of axiom 1.

**Example** **Ⅲ****.4.**Now I consider two random quantities extending(or superposing) on the basis of the same proportion in the system of axiom1.For example,the accurate value for any extension of 2（）1 can not be found due to its next extension 3（）？have not quantities to be selected within the range of the values in the system because the infinitely small is smallest and undivided so that there are not value between 1 and 2. For another instance, ,the accurate value for 4（）3 can aslo not be found due to this value only can be selected in 1 and 2 within the range of the values in the system,so the extension of 4（）3 is meaningless.Therefore Further,that is in a unit of 1、2、3、4……,Corresponding to the ‘include’ values are 1（）1、2（）1、3（）1、4（）1…….For example,the next extension quantities for 2（）1 is 4（）2,again the next is 6（）3,an so on.For another instance, ,the next extension quantities for 5（）1 is 10（）2,again the next is 15（）3,an so on. The ‘include’quantities of the extension of the origin must be any units（）1 due to the extension only can be executed in the way of unit quantities. As shown in figure 2.

**Theorem ****Ⅲ****.5. ** I draw the conclusion that there is no definite values for two random quantities that can not be exactly divided.We let divisor not move(no extension) and allow dividend extend continually to acquire quantities that can exactly divide divisor[3] .It is concluded here that for any two quantities (dividend and divisor )that is not divided exactly we must obtained the aliquot quantites that is integar times of this divisor by the way of extending dividend to infinitely great ,not to infinitesimal.

**Example** **Ⅲ.6. **Seeing figure 3. Now I still consider 4（）3=? in the range of system of axiom1 .No quantities can be choosen for the accurate value (between 1 and 2) calculated due to 1 and 2 can be only selected in the system.To calculate the accurate value for 4（）3,3 is not allowed to extend (motionless),and 4 begins extending,following 7，8，40，400，4000，4000……,it is random for this extention and thus looking for the accurate value for 4（）3 in this system has lost meaning,However, the accurate value for 4000000……（）3 can be found in this system and this value is in the unit of 3 ,that is quantities that can divide exactly by 3.

**Theorem ****Ⅲ****.7. ** It is known fron character 2 that the decimal point is meaningless in the system because the decimal point indicate the quantities that can not be exactly divided.This means that only integer exist and fraction and irrational number do not exist in the system.

**Example** **Ⅲ****.8. **Taking an example, For Fermat’s big theorem xn +yn =zn, since there are no decimal points to exist, its non-integer solutions are meaningless[4], so integer solutions must exist. I will give the exact integer sulutions of the Fermat’s theorem in the next paper.

**Theorem ****Ⅲ****.9. **I draw the conclusion from character 2 that it is meaningless to compare with two quantities that dividend is less than divisor.

**Theorem ****Ⅲ****.10. ** Seeing figure1. The existence of the infinitely small determine that number superposition is in the unit of superposition and this mean that superposition is truncated one.Thus I draw the conclusion that the system of axiom1 is non-continuum.

**Theorem ****Ⅲ****.11. ** A given interval in the system can not be ‘included’ or ‘equal divided’ infinitely and randomly and it is non-existent that the whole quantities can be contained within an given system and for any two given dividend and divisor that is not divided exactly we must obtained the aliquot quantites that is integar times of this divisor by the way of extending dividend to infinitely great ,not to infinitesimal

**Example** **Ⅲ****.12. **For example,in the extent of 20, 20 can only be ‘included’ or ‘equal divided’ by 1,2,5,10. This mean that a given interval is finite quantites,for instance,1 meter long or 300,000 kliometers should be regard as the finite quantites.Taking another example, still considering 4（）3=?, the accurate value for 4（）3 can aslo not be found in the system of axiom1,but the accurate value for 4000000……（）3 can be found in this system and this value is in the unit of 3.Assuming this value is 13333……, then at this moment there are 4000000……/13333……=3,Within the range of 4000000……,one quantity 13333……4 that is beside 13333…… can not be divided exactly by 4000000……,Therefore,the defined value for 4000000……/13333……4 can not be found in the range of 4000000……,you must extend continuously from 4000000…… to more amount quantities to obtain the defined value for the integar times of 13333……4.At the same time, the defined value of the circular constant πcan not be found within an given circumferential lengths ,it can only be found in the quantities more than this given circumferential lengths by the way of extending this given circumferential lengths to infinitely great.As a result,this rule can be executed for random ratio quantities in the system of axiom 1.

**Theorem ****Ⅲ****.13. ** It is inferred from uniqueness of infinitely small and infinitely great and character 6 that the formula for 1（）0= ∞（）1=∞ is non-existent and only ∞（）0=∞ is existent in axiom1. Here‘∞’indicates infinitely great.

**Corollary** **Ⅲ****.14.** It is the sample of axom2 in which the some length of space or time,such as 1 meter long,can be divided infinitely and randomly in the common sense .Meanwhile, it is aslo endowed in this sample by us that the sizes of space and time can be compared (namely there is small and great in sizes) in which axiom 1 and axiom 2 are co-existence(mixed). Some characters can be seen in this mixed axim1 and axiom2, such as the definite values of randomly ‘include’ and ‘equal divided’inside and outside an given system can be found wholly inside this system,decimal point have meaning, and two random quantities can be extended( enlarge and reduce)randomly on the basis of the same proportion,et al.

**Example** **Ⅲ****.15. **For instance, 1.3333…… and 1.3333……4 are meaningful and can be aslo found within one system that can be applied for any decimal and integar quantities within it .Taking an example,any decimal quantities can be included within an given system,such as the range of one meter long or 300,000 kliometers.

**Corollary** **Ⅲ****.16. **Therefore, I draw an conclusion that differential and integral calculus that are continuous calculation of dividend and divisor based on this mixed axim1 and axiom 2 has become not suitable because of existenence of axiom 1[5].