There exist two irrational numbers a and b such that ab is rational. Proof. It i

Question

There exist two irrational numbers a and b such that ab is rational. Proof. It is easy to prove that 2 is irrational. If 2 2 is rational then we are finished (let a = b = 2). If 2 2 is not rational then let a =2 2 and b = 2: then ab = ( 2 2) 2 = 2 2 times 2 = 2 2 and 2 is rational. The logical content of this argument is knowing Psi rightarrow Psi and -Psi rightarrow Psi allows us to conclude Psi. We could include this as a natural deduction style rule: Psi rightarrow Psi -Psi rightarrow Psi/Psi(R)

Explanation / Answer

A rule is sound if and only if
1. The rule is valid.
2. All of its premises are true.

The rule is valid, as I have shown in the other question which you have asked.
All of its premises are true, because in the proof, we show that, irrespective of whether is true or false,  is true. So, by nature of the proof,  ->  and ~ -> 

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