First, if H'(x) = 0, then by definition of the derivative, it must be that H(x) = constant.

If you have F and G s.t. F'=G' = f, then (F - G)' = F' - G' = f - f = 0.

Since (F - G)'(x) = 0, then it must be that (F - G)(x) = constant, or F(x) - G(x) = constant, or F(x) = G(x) + constant.

That is if both F and G belong to the set of anti-derivatives of f, they may differ by a constant.