I'm trying to prove that the product operation on paths induces a well defined operation on path homotopy classes defined by the equation $[f]*[g]=[f*g]$
Let $F$ be a path homotopy between $f$ and $f'$. Then $F(s,0)=f$, $F(s,1)=f'$,
$F(0,t)=x_0$ and $F(1,t)=x_1$
Let $G$ be a path homotopy between $g$ and $g'$. Then $G(s,0)=g$, $G(s,1)=g'$ $G(0,t)=x_1$ and $G(1,t)=x_2$.
I want to show that $[f]*[g]=[f']*[g']$.
Define $H:[0,1]\times [0,1]\to X$ by $H(s,t)=F(2s,t)$ if $s$ in $[0,1/2]$ and
$G(2s-1,t)$ if $s$ in $[1/2,1]$
Then $H(1/2,t)=F(1,t)=x_1=G(0,t)$. Thus $H$ is well defined.
$F$ is cts on $[0,1/2]\times [0,1]$ and $G$ is cts on $[1/2,1]\times [0,1]$ thus $H$ is cts on $[0,1]\times [0,1]$.
Now we have to check the two conditions
$H(s,0),H(s,1),H(0,t),H(1,t)$.
I was able to check $H(0,t)=F(0,t)=x_0$ and $H(1,t)=G(1,t)=x_2$
But I couldn't get the required answer for the other two parts $H(s,0),H(s,1)$. Can somebody please help me with this?