
theorem Th18:
  {p where p is Point of TOP-REAL 2 : p`2 >= 1 - 2 * p`1 } is
  closed Subset of TOP-REAL 2
proof
  reconsider L = {p where p is Point of TOP-REAL 2 : p`2 >= - p`1 } as closed
  Subset of TOP-REAL 2 by JGRAPH_2:47;
  set f = AffineMap (1,0,1/2,-1/2);
  defpred P[Point of TOP-REAL 2] means $1`2 >= 1 - 2 * $1`1;
  {p where p is Point of TOP-REAL 2 : P[p] } is Subset of TOP-REAL 2 from
  JGRAPH_2:sch 1;
  then reconsider
  K = {p where p is Point of TOP-REAL 2 : p`2 >= 1 - 2 * p`1 } as
  Subset of TOP-REAL 2;
  K c= the carrier of TOP-REAL 2;
  then
A1: K c= dom f by FUNCT_2:def 1;
A2: f .: K = L by Th12;
  f is one-to-one by JGRAPH_2:44;
  then K = f " (f .: K) by A1,FUNCT_1:94;
  hence thesis by A2,PRE_TOPC:def 6;
end;
