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) (#) J = G(#)J - H(#)J & J(#)G - J(#)H = J (#) (G - H)
proof
  now
    let n be Element of NAT;
    thus ((G - H) (#) J).n = (G - H).n (#) J.n by Def7
      .= (G.n - H.n) (#) J.n by Th3
      .= G.n (#) J.n - H.n (#) J.n by RFUNCT_1:14
      .= (G(#)J).n - H.n (#) J.n by Def7
      .= (G(#)J).n - (H(#)J).n by Def7
      .= (G(#)J - H(#)J).n by Th3;
  end;
  hence
A1: (G - H) (#) J = G(#)J - H(#)J by FUNCT_2:63;
  thus J(#)G - J(#)H = J(#)G - H(#)J by Th5
    .= (G - H) (#) J by A1,Th5
    .= J (#) (G - H) by Th5;
end;
