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 [;] (d,f)) = H [;] ((h . d),(h * f))

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 [;] (d,f)) = H [;] ((h . d),(h * f))

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 [;] (d,f)) = H [;] ((h . d),(h * f))

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 [;] (d,f)) = H [;] ((h . d),(h * f))

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 [;] (d,f)) = H [;] ((h . d),(h * f))

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 [;] (d,f)) = H [;] ((h . d),(h * f)) )

assume A1: for d1, d2 being Element of D holds h . (F . (d1,d2)) = H . ((h . d1),(h . d2)) ; :: thesis: h * (F [;] (d,f)) = H [;] ((h . d),(h * f))

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 [;] (d,f)) = h * (F .: (g,f)) by FUNCT_2:def 1

.= H .: ((h * g),(h * f)) by A1, Th37

.= H [;] ((h . d),(h * f)) by A2, FUNCOP_1:17 ; :: thesis: verum

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 [;] (d,f)) = H [;] ((h . d),(h * f))

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 [;] (d,f)) = H [;] ((h . d),(h * f))

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 [;] (d,f)) = H [;] ((h . d),(h * f))

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 [;] (d,f)) = H [;] ((h . d),(h * f))

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 [;] (d,f)) = H [;] ((h . d),(h * f))

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 [;] (d,f)) = H [;] ((h . d),(h * f)) )

assume A1: for d1, d2 being Element of D holds h . (F . (d1,d2)) = H . ((h . d1),(h . d2)) ; :: thesis: h * (F [;] (d,f)) = H [;] ((h . d),(h * f))

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 [;] (d,f)) = h * (F .: (g,f)) by FUNCT_2:def 1

.= H .: ((h * g),(h * f)) by A1, Th37

.= H [;] ((h . d),(h * f)) by A2, FUNCOP_1:17 ; :: thesis: verum