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

theorem Th123:
  for cn being Real, K03 being Subset of TOP-REAL 2 st K03={p: p
  `1<=(cn)*(|.p.|) & p`2<=0} holds K03 is closed
proof
  defpred Q[Point of TOP-REAL 2] means $1`2<=0;
  let sn be Real, K003 be Subset of TOP-REAL 2;
  assume
A1: K003={p: p`1<=(sn)*(|.p.|) & p`2<=0};
  reconsider KX={p where p is Point of TOP-REAL 2: Q[p]} as Subset of TOP-REAL
  2 from JGRAPH_2:sch 1;
  defpred P[Point of TOP-REAL 2] means ($1`1<=sn*|.$1.|);
  reconsider K1={p7 where p7 is Point of TOP-REAL 2:P[p7]} as Subset of
  TOP-REAL 2 from JGRAPH_2:sch 1;
A2: {p : P[p] & Q[p]} = {p7 where p7 is Point of TOP-REAL 2:P[p7]} /\ {p1
  where p1 is Point of TOP-REAL 2: Q[p1]} from DOMAIN_1:sch 10;
  K1 is closed & KX is closed by Lm10,JORDAN6:8;
  hence thesis by A1,A2,TOPS_1:8;
end;
