
theorem Th28:
  for C being non empty compact connected Subset of I[01] ex p1,
  p2 being Point of I[01] st p1 <= p2 & C = [. p1, p2 .]
proof
  let C be non empty compact connected Subset of I[01];
  reconsider C9 = C as non empty closed_interval Subset of REAL by Th27;
A1: C9 = [. lower_bound C9, upper_bound C9 .] by INTEGRA1:4;
A2: lower_bound C9 <= upper_bound C9 by Th25;
  then
A3: upper_bound C9 in C by A1,XXREAL_1:1;
  lower_bound C9 in C by A1,A2,XXREAL_1:1;
  then reconsider p1 = lower_bound C9, p2 = upper_bound C9 as Point of I[01]
   by A3;
  take p1, p2;
  thus p1 <= p2 by Th25;
  thus thesis by INTEGRA1:4;
end;
