
theorem Th15:
  for M being non empty MetrSpace, P being non empty Subset of
TopSpaceMetr M, x, y being Point of M holds (dist_min P) . x <= dist (x, y) + (
  dist_min P) . y
proof
  let M be non empty MetrSpace, P be non empty Subset of TopSpaceMetr M, x, y
  be Point of M;
  now
    let z be Point of M;
    assume z in P;
    then (dist_min P) . x <= dist (x, z) by Th13;
    then
A1: dist (x, z) - dist (x, y) >= (dist_min P) . x - dist (x, y) by XREAL_1:13;
    dist (x, z) <= dist (x, y) + dist (y, z) by METRIC_1:4;
    then dist (y, z) >= dist (x, z) - dist (x, y) by XREAL_1:20;
    hence dist (y, z) >= (dist_min P) . x - dist (x, y) by A1,XXREAL_0:2;
  end;
  then (dist_min P) . y >= (dist_min P) . x - dist (x, y) by Th14;
  hence thesis by XREAL_1:20;
end;
