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

theorem
  for A being non empty compact connected Subset of R^1 holds ex a, b
  being Real st a <= b & A = [. a, b .]
proof
  let C be non empty compact connected Subset of R^1;
  reconsider C9 = C as non empty closed_interval Subset of REAL by Th77;
A1: lower_bound C9 <= upper_bound C9 by BORSUK_4:28;
A2: C9 = [. lower_bound C9, upper_bound C9 .] by INTEGRA1:4;
  then
A3: upper_bound C9 in C by A1,XXREAL_1:1;
  lower_bound C9 in C by A2,A1,XXREAL_1:1;
  then reconsider p1 = lower_bound C9, p2 = upper_bound C9 as Point of R^1
   by A3;
  take p1, p2;
  thus p1 <= p2 by BORSUK_4:28;
  thus thesis by INTEGRA1:4;
end;
