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 Th5:
  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 Def7
      .= (H (#) G).n by Def7;
  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 Def7
      .= G.n (#) H.n (#) J.n by Def7
      .= G.n (#) (H.n (#) J.n) by RFUNCT_1:9
      .= G.n (#) (H (#) J).n by Def7
      .= (G (#) (H (#) J)).n by Def7;
  end;
  hence thesis by FUNCT_2:63;
end;
