If it's and not , for all , is a tautology and then is true. (Let be a positive integer,P(a) is the proposition “If n is a positive integer, then n² ≥ n.”
If it's and not , is a positive integer and , therefore is true.
Ok! I'll write instead of
Consider the formula if is false or if is true, then is true.
is true, therefore the implication is true: we've proved
Another way to prove (and even more) is to say that the proposition is a tautology (i.e. is always true). Indeed:
If isn't a positive integer, then is false and the implication is true: is true.
If is a positive integer, because the multiplication on is compatible with the order. So the second part of the implication is true, and again is true.
In conclusion, whatever is (a number, a matrix, a topological space...), is true. In particular, is true.