We define
\(\displaystyle{f:Y+Z\to Y\,,y+z\mapsto y}\) .
Let \(\displaystyle{y+z=y'+z'\in Y+Z}\). Then, \(\displaystyle{yy'=z'z}\) and
\(\displaystyle{yy'\in Y\,,yy'=z'z\in Z}\), so, \(\displaystyle{yy'\in Y\cap Z=\left\{0\right\}}\) and
\(\displaystyle{yy'=0\iff y=y'}\) . (Also, \(\displaystyle{z=z'}\)) .
Therefore, the function \(\displaystyle{f}\) is a well defined map of the \(\displaystyle{\rm{Banach}}\)
space \(\displaystyle{Y+Z}\) to the \(\displaystyle{\rm{Banach}}\) space \(\displaystyle{Y}\) .
It's obvious that \(\displaystyle{f}\) is linear and onto \(\displaystyle{Y}\) . Futhermore,
the graph of \(\displaystyle{f}\)
\(\displaystyle{G_{f}=\left\{\left(y+z,f(y+z)\right): y\in Y\,,z\in Z\right\}\subseteq (Y+Z)\times Y\subseteq X\times X}\)
is a closed subset of \(\displaystyle{(Y+Z)\times Y}\).
According to the closed graph theorem, the function \(\displaystyle{f}\), is bounded, that is
\(\displaystyle{\left(\exists\,C>0\right)\,, f(y+z)=y\leq C\,y+z\,,\forall\,y\in Y\,,\forall\,z\in Z}\) .
