reserve D for non empty set,
  D1,D2,x,y for set,
  n,k for Nat,
  p,x1 ,r for Real,
  f for Function;
reserve F for Functional_Sequence of D1,D2;
reserve G,H,H1,H2,J for Functional_Sequence of D,REAL;

theorem
  G" (#) H" = (G(#)H)"
proof
  now
    let n be Element of NAT;
    thus (G" (#) H").n = (G".n) (#) (H".n) by Def7
      .= ((G.n))^ (#) (H".n) by Def2
      .= ((G.n)^) (#) ((H.n)^) by Def2
      .= (G.n (#) H.n)^ by RFUNCT_1:27
      .= ((G (#) H).n)^ by Def7
      .= ((G (#) H)").n by Def2;
  end;
  hence thesis by FUNCT_2:63;
end;
