Pointwise multiplication and concatenation of loops in a topological group agree up to homotopy

The map $K:[0,1{]}^2\to G$, $K(s,t)=\alpha (s)\beta (t)$, is continuous, and $K(0,0)=K(1,0)=K(0,1)=K(1,1)=e$.

For $u,t\in [0,1]$, put $A_u(t)=\min \{\frac{t}{1-u\mathrm{/}2},1\},B_u(t)=\max \{\frac{t-u\mathrm{/}2}{1-u\mathrm{/}2},0\}\mathrm{.}$ Since $1-u\mathrm{/}2\ge 1\mathrm{/}2$, [L2] and [L3] make these functions continuous on the parameter square. Hence $H_1(t,u)=K(A_u(t),B_u(t))$ is an endpoint-fixed homotopy: at $u=0$ it is $\alpha (t)\beta (t)$, and at $u=1$ it traverses $\alpha$ first and $\beta$ second, so it is $\alpha \ast \beta$ under [F2].

The formula $H_2(t,u)=K(B_u(t),A_u(t))$ is another endpoint-fixed homotopy. At $u=0$ it is again $\alpha (t)\beta (t)$, while at $u=1$ it traverses $\beta$ first and $\alpha$ second, so it is $\beta \ast \alpha$.

If $\alpha$ or $\beta$ is replaced by an endpoint-fixed homotopic loop, multiplying the two homotopies pointwise gives an endpoint-fixed homotopy by [F1] and [L1]. Thus pointwise multiplication is well defined on loop classes, and steps 2.1 and 3.1 identify it with both concatenation orders.