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

theorem
  for A being Subset of R^1, a, b, c being Real st A = [. a, b .[
  \/ ]. b, c .] & a < b & b < c holds Cl A = [. a, c .]
proof
  let A be Subset of R^1, a, b, c be Real;
  assume that
A1: A = [. a, b .[ \/ ]. b, c .] and
A2: a < b and
A3: b < c;
  reconsider B = [. a, b .[, C = ]. b, c .] as Subset of R^1 by TOPMETR:17;
  Cl A = Cl B \/ Cl C by A1,PRE_TOPC:20
    .= [. a, b .] \/ Cl C by A2,Th34
    .= [. a, b .] \/ [. b, c .] by A3,Th35
    .= [. a, c .] by A2,A3,XXREAL_1:174;
  hence thesis;
end;
