Spivak's proof of change of variable theorem.

Question

In proof of change of variable theorem Spivak, on page 69 writes $$\int_A [(f\circ h )\circ g][|\text {det} \ h'| \circ g]|\text {det} \ g'|=\int_A [f\circ (h \circ g)]|\text {det} \ (h\circ g)'|$$ How did we get this step?

Answer 1

First we have $(f\circ h )\circ g=f\circ (h \circ g)$, since composition of functions is associative. Next, by the chain rule $$ (h\circ g)'=(h'\circ g)\,g'. $$ Then $$ \operatorname{det}\bigl((h'\circ g)\,g'\bigr)=\operatorname{det}\bigl((h'\circ g)\bigr)\,\operatorname{det}(g')=\bigl(\operatorname{det}(h')\circ g\bigr)\,\operatorname{det}(g') $$