reserve X for set;

theorem
  for T be non empty TopSpace holds Top InclPoset the topology of T =
  the carrier of T
proof
  let T be non empty TopSpace;
  set L = InclPoset the topology of T, C = the carrier of T;
  the carrier of T = "/\"({},L)
  proof
    reconsider C as Element of L by PRE_TOPC:def 1;
A1: for b being Element of L st b is_<=_than {} holds C >= b
    proof
      let b be Element of L;
      assume b is_<=_than {};
      b in the topology of T;
      hence thesis by Th3;
    end;
    C is_<=_than {};
    hence thesis by A1,YELLOW_0:31;
  end;
  hence thesis by YELLOW_0:def 12;
end;
