
theorem Th14:
  for M being non empty MetrSpace, P being non empty Subset of
  TopSpaceMetr M, r being Real, x being Point of M holds (for y being
  Point of M st y in P holds dist (x,y) >= r) implies (dist_min P) . x >= r
proof
  let M be non empty MetrSpace, P be non empty Subset of TopSpaceMetr M, r be
  Real, x be Point of M;
  consider X being non empty Subset of REAL such that
A1: X = (dist x).:P and
A2: lower_bound ((dist x).:P) = lower_bound X by Th10;
  assume
A3: for y being Point of M st y in P holds dist (x,y) >= r;
  for p being Real st p in X holds p >= r
  proof
    let p be Real;
    assume p in X;
    then consider y being object such that
A4: y in dom dist x and
A5: y in P and
A6: (dist x).y = p by A1,FUNCT_1:def 6;
    reconsider y as Point of M by A4,TOPMETR:12;
    p = dist (x, y) by A6,WEIERSTR:def 4;
    hence thesis by A3,A5;
  end;
  then lower_bound X >= r by SEQ_4:43;
  hence thesis by A2,WEIERSTR:def 6;
end;
