
theorem Th21:
  for R being non empty TopRelStr for A being Subset of R holds
  (for x being Element of R holds downarrow x = Cl {x}) implies
  (A is open implies A is upper)
proof
  let R be non empty TopRelStr, A be Subset of R;
  assume
A1: for x being Element of R holds downarrow x = Cl {x};
  assume
A2: A is open;
  let x,y be Element of R such that
A3: x in A and
A4: x <= y;
  x in downarrow y by A4,WAYBEL_0:17;
  then x in Cl {y} by A1;
  then A meets {y} by A2,A3,PRE_TOPC:24;
  then consider z be object such that
A5: z in A /\ {y} by XBOOLE_0:4;
A6: z in A by A5,XBOOLE_0:def 4;
  z in {y} by A5,XBOOLE_0:def 4;
  hence thesis by A6,TARSKI:def 1;
end;
