let C, D 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 )

reconsider g = C --> e as Function of C,D ;

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

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 )

reconsider g = C --> e as Function of C,D ;

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

now :: thesis: for c being Element of C holds (G [;] (e,f)) . c = g . c

hence
G [;] (e,f) = C --> e
by FUNCT_2:63; :: thesis: verumlet c be Element of C; :: thesis: (G [;] (e,f)) . c = g . c

thus (G [;] (e,f)) . c = G . (e,(f . c)) by FUNCOP_1:53

.= e by A1, Th66

.= g . c ; :: thesis: verum

end;thus (G [;] (e,f)) . c = G . (e,(f . c)) by FUNCOP_1:53

.= e by A1, Th66

.= g . c ; :: thesis: verum