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

theorem Th77:
  for X being non empty compact connected Subset of R^1 holds X
  is non empty non empty closed_interval Subset of REAL
proof
  let C be non empty compact connected Subset of R^1;
  reconsider C9 = C as non empty Subset of REAL by TOPMETR:17;
  C is closed by COMPTS_1:7;
  then
A1: C9 is closed by JORDAN5A:23;
  then
A2: upper_bound C9 in C9 by Th72,RCOMP_1:12;
  C9 is bounded_below bounded_above by Th72;
  then C9 is real-bounded by XXREAL_2:def 11;
  then
A3: lower_bound C9 <= upper_bound C9 by SEQ_4:11;
  lower_bound C9 in C9 by A1,Th72,RCOMP_1:13;
  then [. lower_bound C9, upper_bound C9 .] = C9 by A2,Th74,Th76;
  then C9 is non empty closed_interval by A3,MEASURE5:14;
  hence thesis;
end;
