let D, E be non empty set ; :: thesis: for i being natural Number
for h being Function of D,E
for T1, T2 being Tuple of i,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 natural Number ; :: thesis: for h being Function of D,E
for T1, T2 being Tuple of i,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 Tuple of i,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 Tuple of i,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 A2: i = 0 ; :: thesis: h * (F .: (T1,T2)) = H .: ((h * T1),(h * T2))
then F .: (T1,T2) = <*> D by Lm1;
then A3: h * (F .: (T1,T2)) = <*> E ;
h * T1 = <*> E by A2;
hence h * (F .: (T1,T2)) = H .: ((h * T1),(h * T2)) by ; :: 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 Lm4;
hence h * (F .: (T1,T2)) = H .: ((h * T1),(h * T2)) by ; :: thesis: verum
end;
end;