
theorem Th19:
  for M being non empty MetrSpace, P being non empty Subset of
  TopSpaceMetr M, z being Point of M holds ex w being Point of M st w in P & (
  dist_min P) . z <= dist (w, z)
proof
  let M be non empty MetrSpace, P be non empty Subset of TopSpaceMetr M, z be
  Point of M;
  consider w being object such that
A1: w in P by XBOOLE_0:def 1;
  reconsider w as Point of M by A1,TOPMETR:12;
  take w;
  thus w in P by A1;
  thus thesis by A1,Th13;
end;
