theorem Th25:
  F is having_a_unity implies F"**"(i|->the_unity_wrt F) = the_unity_wrt F
proof
  set e = the_unity_wrt F;
  defpred X[Nat] means F"**"($1|->e) = e;
  assume
A1: F is having_a_unity;
A2: for i st X[i] holds X[i+1]
  proof
    let i;
    assume
A3: F"**"(i|->e) = e;
    thus F"**"((i+1)|->e) = F"**"((i|->e)^<*e*>) by FINSEQ_2:60
      .= F.(F"**"(i|->e),e) by A1,FINSOP_1:4
      .= e by A1,A3,SETWISEO:15;
  end;
  F"**"(0|->e) = F"**" <*>D .= e by A1,FINSOP_1:10;
  then
A4: X[0];
  for i holds X[i] from NAT_1:sch 2(A4,A2);
  hence thesis;
end;
