Given: p is true
prove: p → q is true
assume p and ~q are both true. ~q → r, and r → ~p. therefore, ~p and p cannot be true, so p and ~q cannot be true. therefore, p → q is true.

what type of proof is illustrated above?

a.
proof by contradiction
b.
proof by contraposition
c.
proof by law of detachment
d.
proof by theorem

The
according the statement  
"Assume p and ~q are both true. ~q → r, and r → ~p. Therefore, ~p and p cannot be true, so p and ~q cannot be true. Therefore, p → q is true."
the answerer has just used theorem between combination of ¬p, p, ¬q and q. 

that is called theorem of mathematics logic,

so the proof is  D. proof by theorem