reserve x,y for set;
reserve G for Graph;
reserve vs,vs9 for FinSequence of the carrier of G;
reserve IT for oriented Chain of G;
reserve N for Nat;
reserve n,m,k,i,j for Nat;
reserve r,r1,r2 for Real;
reserve X for non empty set;
reserve p,p1,p2 for Point of TOP-REAL N;
reserve M for non empty MetrSpace;

theorem Th36:
  for P,Q being Subset of TopSpaceMetr(M) st P <> {} & P is
  compact & Q <> {} & Q is compact holds min_dist_min(P,Q)>=0
proof
  let P,Q be Subset of TopSpaceMetr(M);
  assume P <> {} & P is compact & Q <> {} & Q is compact;
  then ex x1,x2 being Point of M st x1 in P & x2 in Q & dist( x1,x2) =
  min_dist_min(P,Q) by WEIERSTR:30;
  hence thesis by METRIC_1:5;
end;
