Ok! I'll write
instead of
We have
Thus
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.