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

theorem Th48:
  for A being Subset of R^1, a being Real st A = ]. a,
  +infty .[ holds Cl A = [. a,+infty .[
proof
  let A be Subset of R^1, a be Real;
  reconsider A9 = A as Subset of R^1;
  reconsider C = [. a,+infty .[ as Subset of R^1 by TOPMETR:17;
  assume
A1: A = ]. a,+infty .[;
  then
A2: A9 is open by Th39;
  C is closed by Th41;
  then
A3: Cl A9 c= C by A1,TOPS_1:5,XXREAL_1:45;
A4: C = A9 \/ {a} by A1,Th42;
  per cases by A4,A3,PRE_TOPC:18,ZFMISC_1:138;
  suppose
    Cl A9 = C;
    hence thesis;
  end;
  suppose
    Cl A9 = A9;
    hence thesis by A1,A2,Th33,Th44;
  end;
end;
