let C, E, D 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)
A2: dom h = D by FUNCT_2:def 1;
A3: dom (h * f) = C by FUNCT_2:def 1;
reconsider g = C --> d as Function of C,D ;
thus h * (F [:] f,d) = h * (F .: f,g) by FUNCT_2:def 1
.= H .: (h * f),(h * g) by A1, Th38
.= H [:] (h * f),(h . d) by A2, A3, FUNCOP_1:23 ; :: thesis: verum