I want to prove that

$\displaystyle a < b \Rightarrow a + c = b + c$

Which of course is not true.

I don't know how to do this kind of proofs.

I mean, when trying to prove

$\displaystyle A \Rightarrow B$

should I try to work on the left side of the implication until A is equal to B? Am I allowed to take on what I have on the left side and make a substitution of that on the right side?

i.e., is this allowed?:

$\displaystyle a < b \Leftrightarrow a + x = b, x > 0$

$\displaystyle a < b \Rightarrow a + c = b + c$

$\displaystyle a + x = b \Rightarrow a + c = b + c$

$\displaystyle a + x = b \Rightarrow a + c = (a + x) + c$