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 + H) = J(#)G + J(#)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 Def5
      .= G.n (#) J.n + H.n (#) J.n by RFUNCT_1:10
      .= (G(#)J).n + H.n (#) J.n by Def7
      .= (G(#)J).n + (H(#)J).n by Def7
      .= (G(#)J + H(#)J).n by Def5;
  end;
  hence (G + H) (#) J = G(#)J + H(#)J by FUNCT_2:63;
  hence J (#) (G + H) = G(#)J + H(#)J by Th5
    .= J(#)G + H(#)J by Th5
    .= J(#)G + J(#)H by Th5;
end;
