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

theorem Th54:
  for A being Subset of R^1, a, b being Real st a < b & A =
  ]. a, b .[ \/ ]. b,+infty .[ holds Cl A = [. a,+infty .[
proof
  let A be Subset of R^1, a, b be Real;
  assume that
A1: a < b and
A2: A = ]. a, b .[ \/ ]. b,+infty .[;
  reconsider B = ]. a, b .[, C = ]. b,+infty .[ as Subset of R^1 by TOPMETR:17;
A3: Cl A = Cl B \/ Cl C by A2,PRE_TOPC:20;
A4: Cl C = [. b,+infty .[ by Th48;
  Cl B = [. a, b .] by A1,Th15;
  hence thesis by A1,A3,A4,Th10;
end;
