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 = H + G & (G + H) + J = G + (H + J)
proof
  now
    let n be Element of NAT;
    thus (G + H).n = H.n + G.n by Def5
      .= (H + G).n by Def5;
  end;
  hence G + H = H + G by FUNCT_2:63;
  now
    let n be Element of NAT;
    thus ((G + H) + J).n = (G + H).n + J.n by Def5
      .= G.n + H.n + J.n by Def5
      .= G.n + (H.n + J.n) by RFUNCT_1:8
      .= G.n + (H + J).n by Def5
      .= (G + (H + J)).n by Def5;
  end;
  hence thesis by FUNCT_2:63;
end;
