To prove an implication “If X, then Y ”, the usual way to do this
is to first assume that X is true, and use this (together with whatever
other facts and hypotheses you have) to deduce Y . This is still a valid
procedure even if X later turns out to be false; the implication does not
guarantee anything about the truth of X, and only guarantees the truth
of Y conditionally on X first being true. For instance, the following is
a valid proof of a true proposition, even though both hypothesis and
conclusion of the proposition are false:
Proposition A.2.2. If 2 + 2 = 5, then 4 = 10 - 4.
Proof. Assume 2+2 = 5. Multiplying both sides by 2, we obtain 4+4 =
10. Subtracting 4 from both sides, we obtain 4 = 10 - 4 as desired.