let C, D be non empty set ; :: thesis: for f being Function of C,D
for F being BinOp of D st F is having_a_unity holds
F [;] ((the_unity_wrt F),f) = f
let f be Function of C,D; :: thesis: for F being BinOp of D st F is having_a_unity holds
F [;] ((the_unity_wrt F),f) = f
let F be BinOp of D; :: thesis: ( F is having_a_unity implies F [;] ((the_unity_wrt F),f) = f )
set e = the_unity_wrt F;
assume A1: F is having_a_unity ; :: thesis: F [;] ((the_unity_wrt F),f) = f

now :: thesis: for c being Element of C holds (F [;] ((the_unity_wrt F),f)) . c = f . c

let c be Element of C; :: thesis: (F [;] ((the_unity_wrt F),f)) . c = f . c
thus (F [;] ((the_unity_wrt F),f)) . c = F . ((the_unity_wrt F),(f . c)) by FUNCOP_1:53
.= f . c by A1, SETWISEO:15 ; :: thesis: verum

end;

hence F [;] ((the_unity_wrt F),f) = f by FUNCT_2:63; :: thesis: verum