reserve i,j,k for Nat,
  r,s,r1,r2,s1,s2,sb,tb for Real,
  x for set,
  GX for non empty TopSpace;
reserve GZ for non empty TopSpace;
reserve f for non constant standard special_circular_sequence,
  G for non empty-yielding Matrix of TOP-REAL 2;

theorem Th15:
  for P being Subset of TOP-REAL 2,s1 st
  P={ |[s,r]| where s,r is Real: s1 <= s }
  holds P is closed
proof
  let P be Subset of TOP-REAL 2, s1;
  reconsider P1= { |[s,r]| where s,r is Real : s < s1 }
  as Subset of TOP-REAL 2 by Lm8;
  assume P= { |[s,r]| where s,r is Real: s >= s1 };
  then
A1: P=P1` by Th10;
  P1 is open by JORDAN1:21;
  hence thesis by A1,TOPS_1:4;
end;
