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 Th12:
  for P being Subset of TOP-REAL 2, s1 st P= { |[r,s]| : s <= s1 }
  holds P is closed
proof
  let P be Subset of TOP-REAL 2, s1;
  reconsider P1= { |[r,s]| where r,s is Real: s > s1 } as Subset of TOP-REAL 2
  by Lm4;
  assume P= { |[r,s]| : s <= s1 };
  then
A1: P=P1` by Th8;
  P1 is open by JORDAN1:22;
  hence thesis by A1,TOPS_1:4;
end;
