
theorem Th3:
  for M being non empty MetrSpace, P being Subset of TopSpaceMetr M
  , x being Point of M st x in P holds 0 in (dist x) .: P
proof
  let M be non empty MetrSpace, P be Subset of TopSpaceMetr M, x be Point of M;
A1: dom dist x = the carrier of TopSpaceMetr M by FUNCT_2:def 1;
  assume x in P;
  then (dist x).x in ((dist x) .: P) by A1,FUNCT_1:def 6;
  hence thesis by Th2;
end;
