reserve k,m,n,p for Nat;
reserve x, a, b, c for Real;
reserve F, f, g, h for Real_Sequence;

theorem Th9:
  for f, g, h being Real_Sequence st g is non-zero holds (f /" g)
  (#) (g /" h) = (f /" h)
proof
  let f, g, h be Real_Sequence;
  set f3 = (f /" g), g3 = (g /" h), h3 = (f /" h);
  assume
A1: g is non-zero;
  for n being Element of NAT holds (f3 (#) g3).n = h3.n
  proof
    let n be Element of NAT;
    set a = f.n, b = (g.n)", c = g.n, d = (h.n)";
A2: g3.n = c * (h".n) by SEQ_1:8
      .= c * d by VALUED_1:10;
A3: h3.n = a * (h".n) by SEQ_1:8
      .= a * d by VALUED_1:10;
A4: g.n <> 0 by A1,SEQ_1:5;
A5: b * c = (1/c) * c .= 1 by A4,XCMPLX_1:106;
    f3.n = a * (g".n) by SEQ_1:8
      .= a * b by VALUED_1:10;
    then (f3 (#) g3).n = (a * b) * (c * d) by A2,SEQ_1:8
      .= ((b * c) * a) * d
      .= h3.n by A3,A5;
    hence thesis;
  end;
  hence thesis by FUNCT_2:63;
end;
