
theorem Th27:
  for C being non empty compact connected Subset of I[01] holds C
  is non empty closed_interval Subset of REAL
proof
  let C be non empty compact connected Subset of I[01];
  reconsider C9 = C as Subset of REAL by BORSUK_1:40,XBOOLE_1:1;
  C9 is bounded_below bounded_above by Th22;
  then
A1: lower_bound C9 <= upper_bound C9 by SEQ_4:11;
A2: C9 is closed by Th24;
  then
A3: upper_bound C9 in C9 by Th22,RCOMP_1:12;
  lower_bound C9 in C9 by A2,Th22,RCOMP_1:13;
  then [. lower_bound C9, upper_bound C9 .] = C9 by A3,Th19,Th26;
  hence thesis by A1,MEASURE5:14;
end;
