Show that composition of paths satisfies the following cancellation property: if $f_0 \cdot g_0 \simeq f_1 \cdot g_1 $ and $g_0 \simeq g_1$, then $f_0 \simeq f_1$.
So I have two homotopies.
So say $g_0,g_1: X \rightarrow Y$ and $f_0,f_1:Y \rightarrow Z$.
Then we know that $G:X \times I \rightarrow Z$ s.t. $G(x,0)=f_0 \cdot g_0(x)$ and $G(x,1)=f_1 \cdot g_1 (x)$. Also, $H:X \times I \rightarrow Y$ s.t. $H(x,0)=g_0(x)$ and $H(x,1)=g_1(x)$.
I was wondering how do you construct the homotopy for f? or is there a simplier way.
I would think you construct a homotopy, but can't see how to.