
theorem Th20:
  for a, b being Real, A being Subset of I[01] st A = [.a,b
  .] holds A is closed
proof
  let a, b be Real, A be Subset of I[01];
  assume
A1: A = [.a,b.];
  per cases;
  suppose
A2: a <= b;
    then
A3: b <= 1 by A1,Th13;
    0 <= a by A1,A2,Th13;
    then
A4: Closed-Interval-TSpace(a,b) is closed SubSpace of
    Closed-Interval-TSpace(0,1) by A2,A3,TREAL_1:3;
    then reconsider
    BA = the carrier of Closed-Interval-TSpace(a,b) as Subset of
    Closed-Interval-TSpace(0,1) by BORSUK_1:1;
    BA is closed by A4,BORSUK_1:def 11;
    hence thesis by A1,A2,TOPMETR:18,20;
  end;
  suppose
    a > b;
    then A = {}I[01] by A1,XXREAL_1:29;
    hence thesis;
  end;
end;
