theorem
  F is commutative implies F.:(T1,T2) = F.:(T2,T1)
proof
  assume
A1: F is commutative;
  per cases;
  suppose
A2: i = 0;
    then F.:(T1,T2) = <*>D by Lm1;
    hence thesis by A2,Lm1;
  end;
  suppose
    i <> 0;
    then reconsider C = Seg i as non empty set;
    T1 is Function of C,D & T2 is Function of C,D by Lm4;
    hence thesis by A1,FUNCOP_1:65;
  end;
end;
