let E, D be non empty set ; :: thesis: for i being Nat
for h being Function of D,E
for T1, T2 being Element of i -tuples_on D
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 .: T1,T2) = H .: (h * T1),(h * T2)
let i be Nat; :: thesis: for h being Function of D,E
for T1, T2 being Element of i -tuples_on D
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 .: T1,T2) = H .: (h * T1),(h * T2)
let h be Function of D,E; :: thesis: for T1, T2 being Element of i -tuples_on D
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 .: T1,T2) = H .: (h * T1),(h * T2)
let T1, T2 be Element of i -tuples_on D; :: 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 .: T1,T2) = H .: (h * T1),(h * T2)
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 .: T1,T2) = H .: (h * T1),(h * T2)
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 .: T1,T2) = H .: (h * T1),(h * T2) )
assume A1: for d1, d2 being Element of D holds h . (F . d1,d2) = H . (h . d1),(h . d2) ; :: thesis: h * (F .: T1,T2) = H .: (h * T1),(h * T2)

per cases ( i = 0 or i <> 0 ) ;

suppose i = 0 ; :: thesis: h * (F .: T1,T2) = H .: (h * T1),(h * T2)

then ( F .: T1,T2 = <*> D & h * T1 = <*> E ) by Lm1, Lm2;
then ( h * (F .: T1,T2) = <*> E & H .: (h * T1),(h * T2) = <*> E ) by FINSEQ_2:87;
hence h * (F .: T1,T2) = H .: (h * T1),(h * T2) ; :: thesis: verum

end;

suppose i <> 0 ; :: thesis: h * (F .: T1,T2) = H .: (h * T1),(h * T2)

then reconsider C = Seg i as non empty set ;
( T1 is Function of C,D & T2 is Function of C,D ) by Lm5;
hence h * (F .: T1,T2) = H .: (h * T1),(h * T2) by A1, Th38; :: thesis: verum

end;

end;