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

theorem Th50:
  for A being Subset of R^1, a being Real st A = ]. -infty,
  a .[ holds Cl A = ]. -infty, a .]
proof
  let A be Subset of R^1, a be Real;
  reconsider A9 = A as Subset of R^1;
  reconsider C = ]. -infty, a .] as Subset of R^1 by TOPMETR:17;
  assume
A1: A = ]. -infty, a .[;
  then
A2: A9 is open by Th39;
  C is closed by Th40;
  then
A3: Cl A9 c= C by A1,TOPS_1:5,XXREAL_1:41;
A4: C = A9 \/ {a} by A1,Th43;
  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,Th47;
  end;
end;
