The quotient of the Path group by the null-homotopic loops is simply connected

Question

I'm trying to understand the short exact sequence for the universal cover given in wiki.

My question is why if $H$ is a path-connected, locally path-connected and semilocally simply connected group, then the quotient $PH/E$ is simply connected.

By $PH$ I denoted the space of paths in $H$ based at the identity together with the compact-open topology (the product of paths is pointwise), $E$ is the normal subgroup of null-homotopic loops.

I'm trying to find a homotopy, but it is difficult to obtain a solution due to the complexity of the compact-open topology...

Is there a direct or quick way to obtain the simple connectedness?