
theorem Th55:
  for C being non empty compact Subset of I[01] st C c= ].0,1.[
holds ex p1, p2 being Point of I[01] st p1 <= p2 & C c= [. p1, p2 .] & [.p1,p2
  .] c= ]. 0, 1 .[
proof
  let C be non empty compact Subset of I[01];
  assume C c= ].0,1.[;
  then consider D being non empty closed_interval Subset of REAL such that
A1: C c= D and
A2: D c= ].0,1.[ and
  lower_bound C = lower_bound D and
  upper_bound C = upper_bound D by Th54;
  consider p1,p2 being Real such that
A3: p1 <= p2 and
A4: D = [.p1,p2.] by MEASURE5:14;
  p1 in D by A3,A4,XXREAL_1:1;
  then
A5: p1 in ].0,1.[ by A2;
  p2 in D by A3,A4,XXREAL_1:1;
  then
A6: p2 in ].0,1.[ by A2;
  ].0,1.[ c= [.0,1.] by XXREAL_1:25;
  then reconsider p1, p2 as Point of I[01] by A5,A6,BORSUK_1:40;
  take p1, p2;
  thus p1 <= p2 by A3;
  thus thesis by A1,A2,A4;
end;
