You already know p → q. You just need to deduce q → q, enabling you to eliminate the disjunction to conclude q as desired. As: p ˅ q, p → q, q → q Ͱ q

1.|  p → q       : premise
2.|_ m → (p ˅ q) : premise
3.|   |_ m       : assumption
4.|   |  p ˅ q   : conditional elimination (2,3)
5.|   |   |_ q   : assumption
6.|   |  q → q   : conditional introduction (5-5)
7.|   |  q       : disjunction elimination (4,1,6)
8.|  m → q       : conditional introduction (3-7)

The proof writer you are using (Stanford?) should allow this, otherwise you might require the premise to be reiterated into the assumptions scope.

You already know p → q. You just need to deduce q → q, enabling you to eliminate the disjunction to conclude q as desired. As: p ˅ q, p → q, q → q Ͱ q

1.|  p → q       : premise
2.|_ m → (p ˅ q) : premise
3.|   |_ m       : assumption
4.|   |  p ˅ q   : conditional elimination (2,3)
5.|   |   |_ q   : assumption
6.|   |  q → q   : conditional introduction (5-5)
7.|   |  q       : disjunction elimination (4,1,6)
8.|  m → q       : conditional introduction (3-7)

The proof writer you are using (Stanford?) should allow this, otherwise you might require the premise to be reiterated into the assumptions scope.

Edit: Ah,I thought this looked familiar.   It is indeed exercise 4-4 of the Standford Logic online course.   There is a "Show Answer" button there, which... does pretty much give the above result.

You already know p → q. You just need to deduce q → q, enabling you to eliminate the disjunction to conclude q as desired. As: p ˅ q, p → q, q → q Ͱ q

1.|  p → q       : premise
2.|_ m → (p ˅ q) : premise
3.|   |_ m       : assumption
4.|   |  p ˅ q   : conditional elimination (2,3)
5.|   |   |_ q   : assumption
6.|   |  q → q   : conditional introduction (5-5)
7.|   |  q       : disjunction elimination (4,1,6)
8.|  m → q       : conditional introduction (3-7)

The proof writer you are using (Staford?) might require you to reiterate the premise into the assumption's scope, but otherwise that is all you need.

You already know p → q. You just need to deduce q → q, enabling you to eliminate the disjunction to conclude q as desired. As: p ˅ q, p → q, q → q Ͱ q

1.|  p → q       : premise
2.|_ m → (p ˅ q) : premise
3.|   |_ m       : assumption
4.|   |  p ˅ q   : conditional elimination (2,3)
5.|   |   |_ q   : assumption
6.|   |  q → q   : conditional introduction (5-5)
7.|   |  q       : disjunction elimination (4,1,6)
8.|  m → q       : conditional introduction (3-7)

The proof writer you are using (Stanford?) should allow this, otherwise you might require the premise to be reiterated into the assumptions scope.