Logic - Given p ⇒ q and m ⇒ p ∨ q, use the Fitch System to prove m ⇒ q.

Question

It seems to be quite simple but I've yet to solve it. There is this, Given p ⇒ q and m ⇒ p ∨ q, how would I prove m ⇒ q? but it does not satisfy the Fitch System I am using since the 7th and 8th step are not possible as assuming q after line 6 would form another subproof.

Answer 1

as assuming q after line 6 would form another subproof.

It is another subproof.   Trivially: assuming $q$, infers $q$.   It is the "first and last line of the second subproof." as stated.

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However, I would use (with reiterations to make it clearer): $$\begin{array}{r|l:l} 1. & p\to q & \text{Premise 1}\\ 2. & m\to (p\vee q) & \text{Premise 2} \\ \hdashline 3. & \quad q & \text{Assumption} \\ \hline 4. & q\to q & 3,3, \text{Implication Introduction} \\ \hdashline 5. & \quad m & \text{Assumption} \\ 6. & \quad m\to (p\vee q) &2,\text{Reiteration} \\ 7. & \quad (p\vee q) & 5,6,\text{Implication Elimination} \\ 8. & \quad p\to q & 1, \text{Reiteration}\\ 9. & \quad q\to q & 4,\text{Reiteration} \\ 10. & \quad q & 7,8,9, \text{Disjunction Elimination} \\ \hline \Box & m\to q & 5,10,\text{Implication Introduction} \end{array}$$