let C, D be non empty set ; :: thesis: for d being Element of D
for f, f9 being Function of C,D
for F being BinOp of D st F is associative holds
(F [:] ((id D),d)) * (F .: (f,f9)) = F .: (f,((F [:] ((id D),d)) * f9))
let d be Element of D; :: thesis: for f, f9 being Function of C,D
for F being BinOp of D st F is associative holds
(F [:] ((id D),d)) * (F .: (f,f9)) = F .: (f,((F [:] ((id D),d)) * f9))
let f, f9 be Function of C,D; :: thesis: for F being BinOp of D st F is associative holds
(F [:] ((id D),d)) * (F .: (f,f9)) = F .: (f,((F [:] ((id D),d)) * f9))
let F be BinOp of D; :: thesis: ( F is associative implies (F [:] ((id D),d)) * (F .: (f,f9)) = F .: (f,((F [:] ((id D),d)) * f9)) )
assume A1: F is associative ; :: thesis: (F [:] ((id D),d)) * (F .: (f,f9)) = F .: (f,((F [:] ((id D),d)) * f9))

now :: thesis: for c being Element of C holds ((F [:] ((id D),d)) * (F .: (f,f9))) . c = (F .: (f,((F [:] ((id D),d)) * f9))) . c

let c be Element of C; :: thesis: ((F [:] ((id D),d)) * (F .: (f,f9))) . c = (F .: (f,((F [:] ((id D),d)) * f9))) . c
thus ((F [:] ((id D),d)) * (F .: (f,f9))) . c = (F [:] ((id D),d)) . ((F .: (f,f9)) . c) by FUNCT_2:15
.= (F [:] ((id D),d)) . (F . ((f . c),(f9 . c))) by FUNCOP_1:37
.= F . (((id D) . (F . ((f . c),(f9 . c)))),d) by FUNCOP_1:48
.= F . ((F . ((f . c),(f9 . c))),d)
.= F . ((f . c),(F . ((f9 . c),d))) by A1
.= F . ((f . c),((F [:] (f9,d)) . c)) by FUNCOP_1:48
.= F . ((f . c),(((F [:] ((id D),d)) * f9) . c)) by FUNCOP_1:50
.= (F .: (f,((F [:] ((id D),d)) * f9))) . c by FUNCOP_1:37 ; :: thesis: verum

end;

hence (F [:] ((id D),d)) * (F .: (f,f9)) = F .: (f,((F [:] ((id D),d)) * f9)) by FUNCT_2:63; :: thesis: verum