On a normed space if $\|u+v\|=\|u\|+\|v\|$ then $\forall s,t \geqslant 0$ we have that $\|su+tv\|=s\|u\|+t\|v\|$
How to prove that $\lVert x + y\rVert = \lVert x\rVert + \lVert y \rVert \implies \lVert tx + (1-t)y\rVert = t\lVert x\rVert + (1-t)\lVert y \rVert$?