reserve X for set,
  Y for non empty set;
reserve n for Nat;
reserve r for Real,
  M for non empty MetrSpace;

theorem Th12:
  for A,B being compact Subset of TopSpaceMetr M st A meets B
  holds min_dist_min(A,B) = 0
proof
  let A,B be compact Subset of TopSpaceMetr M;
  assume A meets B;
  then consider p being object such that
A1: p in A and
A2: p in B by XBOOLE_0:3;
  TopSpaceMetr M = TopStruct (#the carrier of M,Family_open_set M#) by
PCOMPS_1:def 5;
  then reconsider p as Point of M by A1;
  min_dist_min(A,B) >= 0 & min_dist_min(A,B) <= dist(p,p) by A1,A2,Th11,
WEIERSTR:34;
  hence thesis by METRIC_1:1;
end;
