reserve a for Real;
reserve p,q for Point of TOP-REAL 2;

theorem Th62:
  for B0 being Subset of TOP-REAL 2, K0 being Subset of (TOP-REAL
  2)|B0 st B0=NonZero TOP-REAL 2 & K0={p: p`2>=0 & p<>0.TOP-REAL 2} holds K0 is
  closed
proof
  set J0 = NonZero TOP-REAL 2;
  defpred P[Point of TOP-REAL 2] means $1`2>=0;
  set I1 = {p: P[p] & p<>0.TOP-REAL 2};
  let B0 be Subset of TOP-REAL 2,K0 be Subset of (TOP-REAL 2)|B0;
  reconsider K1={p7 where p7 is Point of TOP-REAL 2:P[p7]} as Subset of
  TOP-REAL 2 from JGRAPH_2:sch 1;
A1: I1 = {p7 where p7 is Point of TOP-REAL 2 : P[p7]} /\ J0 from JGRAPH_3:
  sch 2;
  assume B0=J0 & K0=I1;
  then K1 is closed & K0=K1 /\ [#]((TOP-REAL 2)|B0) by A1,JORDAN6:7
,PRE_TOPC:def 5;
  hence thesis by PRE_TOPC:13;
end;
