let C, D, E be non empty set ; :: thesis: for d being Element of D
for f being Function of C,D
for h being Function of D,E
for F being BinOp of D
for H being BinOp of E st ( for d1, d2 being Element of D holds h . (F . (d1,d2)) = H . ((h . d1),(h . d2)) ) holds
h * (F [:] (f,d)) = H [:] ((h * f),(h . d))

let d be Element of D; :: thesis: for f being Function of C,D
for h being Function of D,E
for F being BinOp of D
for H being BinOp of E st ( for d1, d2 being Element of D holds h . (F . (d1,d2)) = H . ((h . d1),(h . d2)) ) holds
h * (F [:] (f,d)) = H [:] ((h * f),(h . d))

let f be Function of C,D; :: thesis: for h being Function of D,E
for F being BinOp of D
for H being BinOp of E st ( for d1, d2 being Element of D holds h . (F . (d1,d2)) = H . ((h . d1),(h . d2)) ) holds
h * (F [:] (f,d)) = H [:] ((h * f),(h . d))

let h be Function of D,E; :: thesis: for F being BinOp of D
for H being BinOp of E st ( for d1, d2 being Element of D holds h . (F . (d1,d2)) = H . ((h . d1),(h . d2)) ) holds
h * (F [:] (f,d)) = H [:] ((h * f),(h . d))

let F be BinOp of D; :: thesis: for H being BinOp of E st ( for d1, d2 being Element of D holds h . (F . (d1,d2)) = H . ((h . d1),(h . d2)) ) holds
h * (F [:] (f,d)) = H [:] ((h * f),(h . d))

let H be BinOp of E; :: thesis: ( ( for d1, d2 being Element of D holds h . (F . (d1,d2)) = H . ((h . d1),(h . d2)) ) implies h * (F [:] (f,d)) = H [:] ((h * f),(h . d)) )
assume A1: for d1, d2 being Element of D holds h . (F . (d1,d2)) = H . ((h . d1),(h . d2)) ; :: thesis: h * (F [:] (f,d)) = H [:] ((h * f),(h . d))
reconsider g = C --> d as Function of C,D ;
A2: ( dom h = D & dom (h * f) = C ) by FUNCT_2:def 1;
thus h * (F [:] (f,d)) = h * (F .: (f,g)) by FUNCT_2:def 1
.= H .: ((h * f),(h * g)) by
.= H [:] ((h * f),(h . d)) by ; :: thesis: verum