
theorem     :: Remark 1.4 (iii)
  for T being Scott up-complete non empty reflexive transitive antisymmetric
  TopRelStr holds T is T_0-TopSpace
proof
  let T be Scott up-complete
  non empty reflexive transitive antisymmetric TopRelStr;
  reconsider T9 = T as Scott TopAugmentation of T by YELLOW_9:44;
  now
    let x,y be Point of T9;
    reconsider x9 = x, y9 = y as Element of T9;
A1: Cl {x9} = downarrow x9 by Th5;
A2: Cl {y9} = downarrow y9 by Th5;
    assume x <> y;
    hence Cl {x} <> Cl {y} by A1,A2,WAYBEL_0:19;
  end;
  hence thesis by TSP_1:def 5;
end;
