page082_Example12

mcguireh

n is odd  if and only if  n² is odd

(#)	Comments	Suppositions and derivations	Theorem and subgoals
		We presuppose the following:
(1)	page077_Example1	If n is odd, then n² is odd
(2)	page080_Example8	If n² is odd, then n is odd

			We start working with the formula being proved as follows:
(3)	page082_Example12		n is odd if and only if n² is odd
			≡ by rewriting the form "`φ₁ if and only if φ₂`" to "`(φ₁ implies φ₂) and (φ₂ implies φ₁)`"
(4)			(n is odd implies n² is odd) and (n² is odd implies n is odd)

			We'll handle formula (4)'s components in separate subparts of this proof, as follows:
			Part 1:
(5)			n is odd implies n² is odd
			↓ by (1)
(6)			true
			That concludes this part of the proof.

			Part 2:
(7)			n² is odd implies n is odd
			↓ by (2)
(8)			true
			That concludes this part of the proof.

			Thus, the theorem that was given is true, by the above partitioning of this proof into subparts.

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