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Because of ~(G0 => G1) iff (G0 /\ ~G1), it suffices to prove
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K G0 => G1
K union {G0 /\ ~G1} F(alse)
"We have to show G0 => G1. Assume G0 /\ ~G1. Then we have ... (derivation of a contradiction)".
Add the hypothesis and the negation of the conclusion to the knowledge and derive a contradiction.