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

theorem
  for A being Subset of R^1, a, b being Real st a < b & A = ].
  -infty, a .[ \/ ]. a, b .[ holds Cl A = ]. -infty, b .]
proof
  let A be Subset of R^1, a, b be Real;
  assume that
A1: a < b and
A2: A = ]. -infty, a .[ \/ ]. a, b .[;
  reconsider B = ]. -infty, a .[, C = ]. a, b .[ as Subset of R^1 by TOPMETR:17
;
A3: Cl C = [. a, b .] by A1,Th15;
  Cl A = Cl B \/ Cl C by A2,PRE_TOPC:20
    .= ]. -infty, a .] \/ [. a, b .] by A3,Th50
    .= ]. -infty, b .] by A1,Th11;
  hence thesis;
end;
