
theorem Th18:
  for T being non empty TopSpace, x, y be Element of InclPoset the
  topology of T holds x "\/" y = x \/ y & x "/\" y = x /\ y
proof
  let T be non empty TopSpace, x, y be Element of InclPoset the topology of T;
A1: the carrier of InclPoset the topology of T = the topology of T by
YELLOW_1:1;
  then x in the topology of T & y in the topology of T;
  then reconsider x9 = x, y9 = y as Subset of T;
  x9 is open & y9 is open by A1,PRE_TOPC:def 2;
  then x9 \/ y9 is open;
  then x9 \/ y9 in the topology of T by PRE_TOPC:def 2;
  hence x "\/" y = x \/ y by YELLOW_1:8;
  x9 /\ y9 in the topology of T by A1,PRE_TOPC:def 1;
  hence thesis by YELLOW_1:9;
end;
