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

theorem
  for A, B being Subset of R^1, a, b, c, d being Real st a < b &
  b <= c & c < d & A = [. a, b .[ & B = ]. c, d .] holds A, B are_separated
proof
  let A, B be Subset of R^1, a, b, c, d be Real;
  assume that
A1: a < b and
A2: b <= c and
A3: c < d and
A4: A = [. a, b .[ and
A5: B = ]. c, d .];
  Cl ]. c, d .] = [. c, d .] by A3,BORSUK_4:11;
  then Cl B = [. c, d .] by A5,JORDAN5A:24;
  then
A6: Cl B misses A by A2,A4,XXREAL_1:95;
  Cl [. a, b .[ = [. a, b .] by A1,BORSUK_4:12;
  then Cl A = [. a, b .] by A4,JORDAN5A:24;
  then Cl A misses B by A2,A5,XXREAL_1:90;
  hence thesis by A6,CONNSP_1:def 1;
end;
