
theorem Th13:
  for M being non empty MetrSpace, P being non empty Subset of
TopSpaceMetr M, x, y being Point of M st y in P holds (dist_min P) . x <= dist
  (x, y)
proof
  let M be non empty MetrSpace, P be non empty Subset of TopSpaceMetr M, x, y
  be Point of M;
A1: dom dist x = the carrier of TopSpaceMetr M & dist (x, y) = (dist x).y by
FUNCT_2:def 1,WEIERSTR:def 4;
  consider X being non empty Subset of REAL such that
A2: X = (dist x) .: P and
A3: lower_bound ((dist x).:P) = lower_bound X by Th10;
  assume y in P;
  then
A4: dist (x, y) in X by A2,A1,FUNCT_1:def 6;
  (dist_min P) . x = lower_bound X & X is bounded_below
   by A2,A3,Th12,WEIERSTR:def 6;
  hence thesis by A4,SEQ_4:def 2;
end;
