let D, C be non empty set ; :: thesis: for e being Element of D
for f being Function of C,D
for F, G being BinOp of D st F is associative & F is having_a_unity & e = the_unity_wrt F & F is having_an_inverseOp & G is_distributive_wrt F holds
G [;] e,f = C --> e
let e be Element of D; :: thesis: for f being Function of C,D
for F, G being BinOp of D st F is associative & F is having_a_unity & e = the_unity_wrt F & F is having_an_inverseOp & G is_distributive_wrt F holds
G [;] e,f = C --> e
let f be Function of C,D; :: thesis: for F, G being BinOp of D st F is associative & F is having_a_unity & e = the_unity_wrt F & F is having_an_inverseOp & G is_distributive_wrt F holds
G [;] e,f = C --> e
let F, G be BinOp of D; :: thesis: ( F is associative & F is having_a_unity & e = the_unity_wrt F & F is having_an_inverseOp & G is_distributive_wrt F implies G [;] e,f = C --> e )
assume A1: ( F is associative & F is having_a_unity & e = the_unity_wrt F & F is having_an_inverseOp & G is_distributive_wrt F ) ; :: thesis: G [;] e,f = C --> e
reconsider g = C --> e as Function of C,D ;

now

let c be Element of C; :: thesis: (G [;] e,f) . c = g . c
thus (G [;] e,f) . c = G . e,(f . c) by FUNCOP_1:66
.= e by A1, Th70
.= g . c by FUNCOP_1:13 ; :: thesis: verum

end;

hence G [;] e,f = C --> e by FUNCT_2:113; :: thesis: verum