I am going through the following lecture notes on lambda calculus by Barendregt and Barendsen :
Here at page 12 , after introducing fixed point theorem a small exercise which is an attempt to prove the following has been posted : $ \exists G \forall X $ $GX =SGX $.
It first proceeds to the implication
$Gx$ = $SGx$
which implies : $G$ = $ \lambda x .SGx $
which implies : $G $ = $ (\lambda gx.Sgx)G$
and then it concludes that :
$G$ = $Y$ $( \lambda g x.Sgx)$ where $Y $ is the fixed point combinator .
I am unable to figure out how was G found out to be $Y$ $( \lambda g x.Sgx)$ in the last step ? How was this conclusion reached up to ?