theorem Th22:
  f1 is commutative & f2 is commutative iff |:f1,f2:| is commutative
proof
  defpred P[set,set] means |:f1,f2:|.($1,$2) = |:f1,f2:|.($2,$1);
  thus f1 is commutative & f2 is commutative implies |:f1,f2:| is commutative
  proof
    assume
A1: for a,b being Element of D1 holds f1.(a,b) = f1.(b,a);
    assume
A2: for a,b being Element of D2 holds f2.(a,b) = f2.(b,a);
A3: for d1,d19 being Element of D1, d2,d29 being Element of D2 holds P[[d1
    ,d2],[d19,d29]]
    proof
      let a1,b1 be Element of D1, a2,b2 be Element of D2;
      thus |:f1,f2:|.([a1,a2],[b1,b2]) = [f1.(a1,b1),f2.(a2,b2)] by Th21
        .= [f1.(b1,a1),f2.(a2,b2)] by A1
        .= [f1.(b1,a1),f2.(b2,a2)] by A2
        .= |:f1,f2:|.([b1,b2],[a1,a2]) by Th21;
    end;
    thus for a,b being Element of [:D1,D2:] holds P[a,b] from AuxCart2( A3);
  end;
  assume
A4: for a,b being Element of [:D1,D2:] holds |:f1,f2:|.(a,b) = |:f1,f2:|
  .(b,a);
  thus for a,b being Element of D1 holds f1.(a,b) = f1.(b,a)
  proof
    set a2 = the Element of D2;
    let a1,b1;
    [f1.(a1,b1),f2.(a2,a2)] = |:f1,f2:|.([a1,a2],[b1,a2]) by Th21
      .= |:f1,f2:|.([b1,a2],[a1,a2]) by A4
      .= [f1.(b1,a1),f2.(a2,a2)] by Th21;
    hence thesis by XTUPLE_0:1;
  end;
  set a1 = the Element of D1;
  let a2,b2;
  [f1.(a1,a1),f2.(a2,b2)] = |:f1,f2:|.([a1,a2],[a1,b2]) by Th21
    .= |:f1,f2:|.([a1,b2],[a1,a2]) by A4
    .= [f1.(a1,a1),f2.(b2,a2)] by Th21;
  hence thesis by XTUPLE_0:1;
end;
