I'm trying to prove that $\deg(fg) = \deg (f) \deg (g)$ for $f:S^1 \to S^1$.
The intermediate step is proving that: if $a$ is a lift of $f \circ \exp$ and $b$ is a lift of $g \circ \exp$ then $a+b$ is a lift of $fg \circ \exp$, however I am having some problems.
To prove that $a+b$ is a lift of $fg \circ \exp$ I surely have to show that $(a+b) \circ \exp = fg \circ \exp$.
However I get that $$(a+b) \circ \exp (t) = a\circ \exp(t) + b\circ\exp(t)$$ $$ = f \circ \exp(t) + g \circ \exp(t)$$ $$ = (f+g)\circ \exp(t)$$
What's going wrong?
