
theorem Th12:
  for M being non empty MetrSpace, P being non empty Subset of
TopSpaceMetr M, x being Point of M, X being Subset of REAL st X = (dist x) .: P
  holds X is bounded_below
proof
  let M be non empty MetrSpace, P be non empty Subset of TopSpaceMetr M, x be
  Point of M, X be Subset of REAL;
  assume
A1: X = (dist x).:P;
  take 0;
  let y be ExtReal;
  thus y in X implies 0 <= y by A1,Th4;
end;
