Remmert page 15 chapter 0 it says that angle preserving mapping is R-linear and injective.
We want to prove:
Given $T:\mathbb{C}\rightarrow \mathbb{C}$ a $\mathbb{R}$ linear map which satisfies $|z||w|\langle T(z),T(w)\rangle = |T(z)||T(w)|\langle z,w\rangle$ that isn't a 0 map. Then it is injective.
There exist equivalent definitions, which is not yet introduced but we don't want cheat.
For injectivity we need: $T(z)=T(w) \Rightarrow z=w $ for all $z,w \in \mathbb{C} $
But then, how to continue? Please, do tell.