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Line 30: − Conditional sub-proof of indirect proof: Let's introduce a second hypothesis and see what follows from it.+ − − :Simple assumption: One or more sets of ''two'' natural numbers, a pair ''n''<sub>1</sub> and ''n''<sub>2</sub>, for example, exist that have a positive integral factor ''f'' apart from one and themselves and common to both of them. − − :Corollary to assumption: Through the definition of prime numbers such as described in the first "conceptualization" section, if any pair of positive integers ''n''<sub>1</sub> and ''n''<sub>2</sub> have an common integral factor ''f'' apart from one and themselves, they are not prime because they ''both'' have a non-prime type of positive integral factor; in fact it's the same one. − − Conditional sub-proof to be proven: ''Assuming no pair of prime numbers can possess the qualities of the pair set with the described factor, no three lines having prime numbers as lengths can form a right triangle.'' Line 52:
Line 46: − + − − 4. Because of this second assumption, the square of the prime legs ''l''<sub>1</sub><sup>2</sup> and ''l''<sub>2</sub><sup>2</sup> only have a prime factor doubled which must be less than prime factor ''h'' and therefore the prime hypoteneuse is not divisible by the squared legs. − 5. Therefore the sum of each of the squared legs divided by the hypoteneuse can never be a whole number like the ''h'' that remains when ''h'' is divided through.+
→Corollary: No three lines with prime numbers as lengths can form a right triangle: replaced first simple assumption with second, located definition of prime in proof.
===Second hypothesis===
===Second hypothesis===
Simple assumption: The hypoteneuse is always larger than either of its lengths.
That is, it is to be proven that the first proposition (that is, the proposition that is the simple assumption, the ''assumption for a conditional proof'') leads to the second.
That is, it is to be proven that the first proposition (that is, the proposition that is the simple assumption, the ''assumption for a conditional proof'') leads to the second.
2. And since ''l''<sub>1</sub> and ''l''<sub>2</sub> are whole numbers, that which results when they are multiplied by themselves, namely ''l''<sub>1</sub><sup>2</sup> and ''l''<sub>2</sub><sup>2</sup>, certainly are as well.
2. And since ''l''<sub>1</sub> and ''l''<sub>2</sub> are whole numbers, that which results when they are multiplied by themselves, namely ''l''<sub>1</sub><sup>2</sup> and ''l''<sub>2</sub><sup>2</sup>, certainly are as well.
3. 2nd simple assumption: The hypoteneuse is always larger than either of its lengths
3. Because of the simple assumption, the square of the prime legs ''l''<sub>1</sub><sup>2</sup> and ''l''<sub>2</sub><sup>2</sup> only have a prime factor doubled which must be less than prime factor ''h'' and therefore the leg lengths squared are not divisible by the prime hypoteneuse by the given definition of a prime number.
4. Therefore the sum of each of the squared legs divided by the hypoteneuse can never be a whole number like the ''h'' that remains when ''h'' is divided through.
[[Category:Plane Geometry]]
[[Category:Plane Geometry]]
[[Category:Mathematics]]
[[Category:Mathematics]]
[[Category:Geometric Theorems]]
[[Category:Geometric Theorems]]