Prove there is a unique linear map $f:U\bigoplus V\rightarrow U+V$
Proof:
The only choice is $f(u,v)=u+v$ because if $f$ were linear, $f(u,v)=f((u,0)+(0,v))=f(u,0)+f(0,v)$. An then show this is actually linear.
Q.E.D.
But I don't get why the first line shows uniqueness? Why can't there be another $g(u,v)=g(u,0)+g(0,v)$