
theorem
  for M being non empty MetrSpace holds for P being Subset of
  TopSpaceMetr(M) st P <> {} & P is compact holds for p1,p2 being Point of M
holds for x1,x2 being Real holds x1 = (dist_min(P)).p1 & x2 = (dist_min(P)).p2
  implies |.x1 - x2.| <= dist(p1,p2)
proof
  let M be non empty MetrSpace;
  let P be Subset of TopSpaceMetr(M);
  assume
A1: P <> {} & P is compact;
  let p1,p2 be Point of M;
  let x1,x2 be Real;
  assume
A2: x1 = (dist_min(P)).p1 & x2 = (dist_min(P)).p2;
  (dist_min(P)).p1 = lower_bound((dist(p1)).:P) & (dist_min(P)).p2 =
  lower_bound((dist(p2)).:P) by Def6;
  hence thesis by A1,A2,Th22;
end;
