Let $h: I \times I \rightarrow X$ be a continuous function, and let $a, b, c, d$ be the paths in $X$ defined as follows:
Then I want to prove that $a.b$ is path homotopic to $c.d$.
I tried to write homotopy explicitly but things got messy. The idea is that treating $a.b$ and $c.d$ as paths and homotope them to the diagonal of the square. Does this idea work?