theorem
  F is having_a_unity implies
  F.:(i|->the_unity_wrt F,T) = T & F.:(T,i|->the_unity_wrt F) = T
proof
  assume
A1: F is having_a_unity;
  per cases;
  suppose
A2: i = 0;
    then T = <*>D;
    hence thesis by A2,Lm1;
  end;
  suppose
    i <> 0;
    then reconsider C = Seg i as non empty set;
    T is Function of C,D by Lm4;
    hence thesis by A1,Th43;
  end;
end;
