
theorem Th21:
  for p1, p2 being Point of I[01] st p1 <= p2 holds [. p1, p2 .]
  is non empty compact connected Subset of I[01]
proof
  let p1, p2 be Point of I[01];
A1: p2 <= 1 by BORSUK_1:43;
  set S = [. p1, p2 .];
  reconsider S as Subset of I[01] by BORSUK_1:40,XXREAL_2:def 12;
  assume
A2: p1 <= p2;
  then
A3: Closed-Interval-TSpace(p1,p2) is connected by TREAL_1:20;
A4: S is closed by Th20;
  0 <= p1 by BORSUK_1:43;
  then I[01] | S = Closed-Interval-TSpace(p1,p2) by A2,A1,TOPMETR:24;
  hence thesis by A4,A3,COMPTS_1:8,CONNSP_1:def 3;
end;
