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Prove by cases, First case is A then from A prove A v C using v Intro(A) . Case 2 is B. You will need another set of cases in Case 2, namely those of your other premise ~B (not B) v C. So case 2 (B) Will have two sub proofs. The first of the sub proofs is ~B which you prove by using _|_ Intro citing B and ~B. After this you can use _|_ Elim to prove whatever you like. Since we like or want A v C you should then add one more step and prove from _|_ A v C (using _|_ Elim). The second sub proof within case 2 should be C (this can be figured out again by looking at what you haven't used from your premises, C is the only thing you haven't used). So under C make a new line and prove A v C using v Intro. Prove _|_ for Case 2 by using vElim and then Prove A v C for your final proof by v ELim from your two cases (A B).

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