let D, C 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 .: (C --> (the_unity_wrt F)),f = f & F .: f,(C --> (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 .: (C --> (the_unity_wrt F)),f = f & F .: f,(C --> (the_unity_wrt F)) = f )
let F be BinOp of D; :: thesis: ( F is having_a_unity implies ( F .: (C --> (the_unity_wrt F)),f = f & F .: f,(C --> (the_unity_wrt F)) = f ) )
set e = the_unity_wrt F;
reconsider g = C --> (the_unity_wrt F) as Function of C,D ;
assume A1: F is having_a_unity ; :: thesis: ( F .: (C --> (the_unity_wrt F)),f = f & F .: f,(C --> (the_unity_wrt F)) = f )

now

let c be Element of C; :: thesis: (F .: g,f) . c = f . c
thus (F .: g,f) . c = F . (g . c),(f . c) by FUNCOP_1:48
.= F . (the_unity_wrt F),(f . c) by FUNCOP_1:13
.= f . c by A1, SETWISEO:23 ; :: thesis: verum

end;

hence F .: (C --> (the_unity_wrt F)),f = f by FUNCT_2:113; :: thesis: F .: f,(C --> (the_unity_wrt F)) = f

now

let c be Element of C; :: thesis: (F .: f,g) . c = f . c
thus (F .: f,g) . c = F . (f . c),(g . c) by FUNCOP_1:48
.= F . (f . c),(the_unity_wrt F) by FUNCOP_1:13
.= f . c by A1, SETWISEO:23 ; :: thesis: verum

end;

hence F .: f,(C --> (the_unity_wrt F)) = f by FUNCT_2:113; :: thesis: verum