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→Reasoning
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. 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 (otherwise it would give it a factor both greater than one and itself).
3. If two primes can have no common factor, squaring the primes add no new factor, so their squares don't either.
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, contradicting the rephrased original premise of the indirect proof and demonstrating the opposite of the assumption to be true, namely the three lengths in a pythagorean triangle can never be all prime.
4. 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 (otherwise it would give it a factor both greater than one and itself).
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, contradicting the rephrased original premise of the indirect proof and demonstrating the opposite of the assumption to be true, namely the three lengths in a pythagorean triangle can never be all prime.
[[Category:Plane Geometry]]
[[Category:Plane Geometry]]
[[Category:Mathematics]]
[[Category:Mathematics]]
[[Category:Geometric Theorems]]
[[Category:Geometric Theorems]]