let C, D be non empty set ; for B being Element of Fin C
for F being BinOp of D
for f, f9 being Function of C,D st F is commutative & F is associative & F is having_a_unity holds
F . ((F $$ (B,f)),(F $$ (B,f9))) = F $$ (B,(F .: (f,f9)))
let B be Element of Fin C; for F being BinOp of D
for f, f9 being Function of C,D st F is commutative & F is associative & F is having_a_unity holds
F . ((F $$ (B,f)),(F $$ (B,f9))) = F $$ (B,(F .: (f,f9)))
let F be BinOp of D; for f, f9 being Function of C,D st F is commutative & F is associative & F is having_a_unity holds
F . ((F $$ (B,f)),(F $$ (B,f9))) = F $$ (B,(F .: (f,f9)))
let f, f9 be Function of C,D; ( F is commutative & F is associative & F is having_a_unity implies F . ((F $$ (B,f)),(F $$ (B,f9))) = F $$ (B,(F .: (f,f9))) )
set e = the_unity_wrt F;
assume A1:
( F is commutative & F is associative & F is having_a_unity )
; F . ((F $$ (B,f)),(F $$ (B,f9))) = F $$ (B,(F .: (f,f9)))
then
( F . ((the_unity_wrt F),(the_unity_wrt F)) = the_unity_wrt F & ( for d1, d2, d3, d4 being Element of D holds F . ((F . (d1,d2)),(F . (d3,d4))) = F . ((F . (d1,d3)),(F . (d2,d4))) ) )
by Lm3, SETWISEO:15;
hence
F . ((F $$ (B,f)),(F $$ (B,f9))) = F $$ (B,(F .: (f,f9)))
by A1, Th9; verum