
theorem Th12:
  for A being Subset of I[01], a, b being Real st a < b & A
  = [.a,b.[ holds Cl A = [.a,b.]
proof
  let A be Subset of I[01], a, b be Real;
  assume that
A1: a < b and
A2: A = [.a,b.[;
  reconsider A1 = [.a,b.[ as Subset of R^1 by TOPMETR:17;
A3: Cl [.a,b.[ = [.a,b.] by A1,Th9;
  Cl A = (Cl A1) /\ [#]I[01] by A2,PRE_TOPC:17
    .= [.a,b.] /\ [#]I[01] by A3,JORDAN5A:24
    .= [.a,b.] by A1,A2,Th7,XBOOLE_1:28;
  hence thesis;
end;
