let E, D be non empty set ; :: thesis: for d being Element of D
for i being Nat
for h being Function of D,E
for T 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 [;] d,T) = H [;] (h . d),(h * T)
let d be Element of D; :: thesis: for i being Nat
for h being Function of D,E
for T 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 [;] d,T) = H [;] (h . d),(h * T)
let i be Nat; :: thesis: for h being Function of D,E
for T 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 [;] d,T) = H [;] (h . d),(h * T)
let h be Function of D,E; :: thesis: for T 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 [;] d,T) = H [;] (h . d),(h * T)
let T 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 [;] d,T) = H [;] (h . d),(h * T)
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,T) = H [;] (h . d),(h * T)
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,T) = H [;] (h . d),(h * T) )
assume A1: for d1, d2 being Element of D holds h . (F . d1,d2) = H . (h . d1),(h . d2) ; :: thesis: h * (F [;] d,T) = H [;] (h . d),(h * T)

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

suppose i = 0 ; :: thesis: h * (F [;] d,T) = H [;] (h . d),(h * T)

then ( F [;] d,T = <*> D & h * T = <*> E ) by Lm1, Lm3;
then ( h * (F [;] d,T) = <*> E & H [;] (h . d),(h * T) = <*> E ) by FINSEQ_2:93;
hence h * (F [;] d,T) = H [;] (h . d),(h * T) ; :: thesis: verum

end;

suppose i <> 0 ; :: thesis: h * (F [;] d,T) = H [;] (h . d),(h * T)

then reconsider C = Seg i as non empty set ;
T is Function of C,D by Lm5;
hence h * (F [;] d,T) = H [;] (h . d),(h * T) by A1, Th39; :: thesis: verum

end;

end;