This is a question for my philosophy.
Prove this valid using any of the rules we've studied so far:
A v (B & C)
(A v C) > ~(G & O) / ~G v ~O
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The following proof uses these rules: disjunction introduction (∨I), conjunction elimination (∧E), disjunction elimination (∨E), conditional elimination (→E) and the DeMorgan Rule (DeM).
Of the rules used, the DeMorgan Rule may not be in the "rules we've studied so far".
Here is another proof not using the DeMorgan Rule. It is based on the derivation for the DeMorgan Rule in forall x: Calgary Remix, page 142. It uses the following additional rules: contradiction introduction (⊥I), negation introduction (¬I) and the law of excluded middle (LEM).
P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/