let C, D be non empty set ; :: thesis: for B being Element of Fin C

for F, G being BinOp of D

for f, f9 being Function of C,D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G = F * ((id D),(the_inverseOp_wrt F)) holds

G . ((F $$ (B,f)),(F $$ (B,f9))) = F $$ (B,(G .: (f,f9)))

let B be Element of Fin C; :: thesis: for F, G being BinOp of D

for f, f9 being Function of C,D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G = F * ((id D),(the_inverseOp_wrt F)) holds

G . ((F $$ (B,f)),(F $$ (B,f9))) = F $$ (B,(G .: (f,f9)))

let F, G be BinOp of D; :: thesis: for f, f9 being Function of C,D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G = F * ((id D),(the_inverseOp_wrt F)) holds

G . ((F $$ (B,f)),(F $$ (B,f9))) = F $$ (B,(G .: (f,f9)))

let f, f9 be Function of C,D; :: thesis: ( F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G = F * ((id D),(the_inverseOp_wrt F)) implies G . ((F $$ (B,f)),(F $$ (B,f9))) = F $$ (B,(G .: (f,f9))) )

assume that

A1: ( F is commutative & F is associative & F is having_a_unity ) and

A2: ( F is having_an_inverseOp & G = F * ((id D),(the_inverseOp_wrt F)) ) ; :: thesis: G . ((F $$ (B,f)),(F $$ (B,f9))) = F $$ (B,(G .: (f,f9)))

set e = the_unity_wrt F;

( G . ((the_unity_wrt F),(the_unity_wrt F)) = the_unity_wrt F & ( for d1, d2, d3, d4 being Element of D holds F . ((G . (d1,d2)),(G . (d3,d4))) = G . ((F . (d1,d3)),(F . (d2,d4))) ) ) by A1, A2, FINSEQOP:86, FINSEQOP:89;

hence G . ((F $$ (B,f)),(F $$ (B,f9))) = F $$ (B,(G .: (f,f9))) by A1, Th9; :: thesis: verum

for F, G being BinOp of D

for f, f9 being Function of C,D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G = F * ((id D),(the_inverseOp_wrt F)) holds

G . ((F $$ (B,f)),(F $$ (B,f9))) = F $$ (B,(G .: (f,f9)))

let B be Element of Fin C; :: thesis: for F, G being BinOp of D

for f, f9 being Function of C,D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G = F * ((id D),(the_inverseOp_wrt F)) holds

G . ((F $$ (B,f)),(F $$ (B,f9))) = F $$ (B,(G .: (f,f9)))

let F, G be BinOp of D; :: thesis: for f, f9 being Function of C,D st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G = F * ((id D),(the_inverseOp_wrt F)) holds

G . ((F $$ (B,f)),(F $$ (B,f9))) = F $$ (B,(G .: (f,f9)))

let f, f9 be Function of C,D; :: thesis: ( F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G = F * ((id D),(the_inverseOp_wrt F)) implies G . ((F $$ (B,f)),(F $$ (B,f9))) = F $$ (B,(G .: (f,f9))) )

assume that

A1: ( F is commutative & F is associative & F is having_a_unity ) and

A2: ( F is having_an_inverseOp & G = F * ((id D),(the_inverseOp_wrt F)) ) ; :: thesis: G . ((F $$ (B,f)),(F $$ (B,f9))) = F $$ (B,(G .: (f,f9)))

set e = the_unity_wrt F;

( G . ((the_unity_wrt F),(the_unity_wrt F)) = the_unity_wrt F & ( for d1, d2, d3, d4 being Element of D holds F . ((G . (d1,d2)),(G . (d3,d4))) = G . ((F . (d1,d3)),(F . (d2,d4))) ) ) by A1, A2, FINSEQOP:86, FINSEQOP:89;

hence G . ((F $$ (B,f)),(F $$ (B,f9))) = F $$ (B,(G .: (f,f9))) by A1, Th9; :: thesis: verum