let C, D be non empty set ; :: thesis: for B being Element of Fin C
for F being BinOp of D
for u being UnOp of D
for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & u . (the_unity_wrt F) = the_unity_wrt F & u is_distributive_wrt F holds
u . (F $$ (B,f)) = F $$ (B,(u * f))
let B be Element of Fin C; :: thesis: for F being BinOp of D
for u being UnOp of D
for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & u . (the_unity_wrt F) = the_unity_wrt F & u is_distributive_wrt F holds
u . (F $$ (B,f)) = F $$ (B,(u * f))
let F be BinOp of D; :: thesis: for u being UnOp of D
for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & u . (the_unity_wrt F) = the_unity_wrt F & u is_distributive_wrt F holds
u . (F $$ (B,f)) = F $$ (B,(u * f))
let u be UnOp of D; :: thesis: for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & u . (the_unity_wrt F) = the_unity_wrt F & u is_distributive_wrt F holds
u . (F $$ (B,f)) = F $$ (B,(u * f))
let f be Function of C,D; :: thesis: ( F is commutative & F is associative & F is having_a_unity & u . (the_unity_wrt F) = the_unity_wrt F & u is_distributive_wrt F implies u . (F $$ (B,f)) = F $$ (B,(u * f)) )
assume A1: ( F is commutative & F is associative & F is having_a_unity & u . (the_unity_wrt F) = the_unity_wrt F ) ; :: thesis: ( not u is_distributive_wrt F or u . (F $$ (B,f)) = F $$ (B,(u * f)) )
assume for d1, d2 being Element of D holds u . (F . (d1,d2)) = F . ((u . d1),(u . d2)) ; :: according to BINOP_1:def 20 :: thesis: u . (F $$ (B,f)) = F $$ (B,(u * f))
hence u . (F $$ (B,f)) = F $$ (B,(u * f)) by A1, Th18; :: thesis: verum