let C, D be non empty set ; :: thesis: for d being Element of D
for f being Function of C,D
for F being BinOp of D
for u being UnOp of D st u is_distributive_wrt F holds
u * (F [:] (f,d)) = F [:] ((u * f),(u . d))
let d be Element of D; :: thesis: for f being Function of C,D
for F being BinOp of D
for u being UnOp of D st u is_distributive_wrt F holds
u * (F [:] (f,d)) = F [:] ((u * f),(u . d))
let f be Function of C,D; :: thesis: for F being BinOp of D
for u being UnOp of D st u is_distributive_wrt F holds
u * (F [:] (f,d)) = F [:] ((u * f),(u . d))
let F be BinOp of D; :: thesis: for u being UnOp of D st u is_distributive_wrt F holds
u * (F [:] (f,d)) = F [:] ((u * f),(u . d))
let u be UnOp of D; :: thesis: ( u is_distributive_wrt F implies u * (F [:] (f,d)) = F [:] ((u * f),(u . d)) )
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 [:] (f,d)) = F [:] ((u * f),(u . d))
hence u * (F [:] (f,d)) = F [:] ((u * f),(u . d)) by Th40; :: thesis: verum