Proof: Make auxiliary line AC in the drawing.
Because ABCD is a square, CF is the bisector of ∠DCG outside the square.
So AC⊥CF and ∠ ECF = 45, which means that EC is the bisector of ∠ACF.
Because AE⊥EF, EC is also the bisector of ∠AEF.
So ∠ ECF = ∠ CEF = 45.
∠ EFC = 90, then EF=FC.
AE⊥EF
AC⊥CF
So AEFC is square, which means AE=EF.