Let $E, F$ be projections in a von Neumann algebra such that $E ∼ E^⊥$ and $F ∼ F^⊥$. Show that $E ∼ F$.
This is an exercise in J B Conway's A Course in Operator Theory. Here $E ∼ F$ if there exists a partial isometry $V$ in that von Neumann algebra such that $V^*V = E$ and $VV^* = F$. I tried using a theorem there which states if $E, F$ are projections in a von Neumann algebra, then there exists a central projection $Z$ such that $EZ \preccurlyeq FZ$ and $FZ^⊥ \preccurlyeq EZ^⊥$ (Here $A \preccurlyeq B$ if there exists a projection $C$ in that von Neumann algebra such that $C \leq B$ and $A ∼ C$). My aim was to show $E \preccurlyeq F$ and $F \preccurlyeq E$, then $E ∼ F$ follows from another theorem.
Please get me started. Any help is appreciated.