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

now

let c be Element of C; :: thesis: (F [:] f,(the_unity_wrt F)) . c = f . c
thus (F [:] f,(the_unity_wrt F)) . c = F . (f . c),(the_unity_wrt F) by FUNCOP_1:60
.= f . c by A1, SETWISEO:23 ; :: thesis: verum

end;

hence F [:] f,(the_unity_wrt F) = f by FUNCT_2:113; :: thesis: verum