
theorem Th4:
  for M being non empty MetrSpace, P being Subset of TopSpaceMetr M
  , x being Point of M, y being Real st y in (dist x) .: P holds y >= 0
proof
  let M be non empty MetrSpace, P be Subset of TopSpaceMetr M, x be Point of M
  , y be Real;
  assume y in (dist x) .: P;
  then consider z being object such that
  z in dom dist x and
A1: z in P and
A2: y = (dist x).z by FUNCT_1:def 6;
  reconsider z9 = z as Point of M by A1,TOPMETR:12;
  y = dist (x, z9) by A2,WEIERSTR:def 4;
  hence thesis by METRIC_1:5;
end;
