reserve x1, x2, x3, x4, x5, x6, x7 for set;

theorem Th74:
  for C being non empty compact connected Subset of R^1, C9 being
  Subset of REAL st C = C9 & [. lower_bound C9, upper_bound C9 .] c= C9
   holds [. lower_bound C9, upper_bound C9.] = C9
proof
  let C be non empty compact connected Subset of R^1, C9 be Subset of REAL;
  assume that
A1: C = C9 and
A2: [. lower_bound C9, upper_bound C9 .] c= C9;
  assume [. lower_bound C9, upper_bound C9 .] <> C9;
  then not C9 c= [. lower_bound C9, upper_bound C9 .] by A2,XBOOLE_0:def 10;
  then consider c being object such that
A3: c in C9 and
A4: not c in [. lower_bound C9, upper_bound C9 .];
  reconsider c as Real by A3;
A5: c <= upper_bound C9 by A1,A3,Th73;
  lower_bound C9 <= c by A1,A3,Th73;
  hence thesis by A4,A5,XXREAL_1:1;
end;
