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

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

suppose A2: i = 0 ; :: thesis: h * (F [:] (T,d)) = H [:] ((h * T),(h . d))

then F [:] (T,d) = <*> D by Lm3;
then A3: h * (F [:] (T,d)) = <*> E ;
h * T = <*> E by A2;
hence h * (F [:] (T,d)) = H [:] ((h * T),(h . d)) by A3, FINSEQ_2:85; :: thesis: verum

end;

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

then reconsider C = Seg i as non empty set ;
T is Function of C,D by Lm4;
hence h * (F [:] (T,d)) = H [:] ((h * T),(h . d)) by A1, Th39; :: thesis: verum

end;

end;