You are being too literal -- I was providing an analogy, not an example.
Also:
> ... all theorems about a certain statement being true must be reducible to a single irreducible logical chain of argument.
Citation needed ... I have no reason to believe this is true.
But here's an example of two proofs.
Proving sqrt(2) is irrational.
Proof 1:
Every rational number has a finite Continued Fraction representation. But the normalised Continued Fraction representation of sqrt(2) is [1;2,2,2,...], which is infinite. Since this is infinite, sqrt(2) is irrational.
Proof 2:
Consider integers a and b, and suppose 2(b²)=a². Consider the prime decompositions of a and b, and count how many times "2" turns up on each side. It's odd on the left, it's even on the right, so this cannot happen. Therefore we can never have integers a and b with 2(b²)=a². Therefore we can't have 2=(a²)/(b²)=(a/b)². So any fraction when squared cannot equal 2, so sqrt(2) is irrational.
Also:
> ... all theorems about a certain statement being true must be reducible to a single irreducible logical chain of argument.
Citation needed ... I have no reason to believe this is true.
But here's an example of two proofs.
Proving sqrt(2) is irrational.
Proof 1: Every rational number has a finite Continued Fraction representation. But the normalised Continued Fraction representation of sqrt(2) is [1;2,2,2,...], which is infinite. Since this is infinite, sqrt(2) is irrational.
Proof 2: Consider integers a and b, and suppose 2(b²)=a². Consider the prime decompositions of a and b, and count how many times "2" turns up on each side. It's odd on the left, it's even on the right, so this cannot happen. Therefore we can never have integers a and b with 2(b²)=a². Therefore we can't have 2=(a²)/(b²)=(a/b)². So any fraction when squared cannot equal 2, so sqrt(2) is irrational.
Do these really feel "the same" to you?