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 Th37:
  for P,Q being Subset of TopSpaceMetr(M) st P <> {} & P is
  compact & Q <> {} & Q is compact holds P misses Q iff min_dist_min(P,Q)>0
proof
  let P,Q be Subset of TopSpaceMetr(M);
  assume that
A1: P <> {} and
A2: P is compact and
A3: Q <> {} and
A4: Q is compact;
A5: now
    set p = the Element of P /\ Q;
    assume
A6: P /\ Q <>{};
    then
A7: p in P by XBOOLE_0:def 4;
    then reconsider p9=p as Element of TopSpaceMetr(M);
    reconsider q=p9 as Point of M by TOPMETR:12;
    the distance of M is Reflexive by METRIC_1:def 6;
    then (the distance of M).(q,q)=0 by METRIC_1:def 2;
    then
A8: dist(q,q)=0 by METRIC_1:def 1;
    p in Q by A6,XBOOLE_0:def 4;
    hence not min_dist_min(P,Q)>0 by A2,A4,A7,A8,WEIERSTR:34;
  end;
  consider x1,x2 being Point of M such that
A9: x1 in P & x2 in Q and
A10: dist(x1,x2) = min_dist_min(P,Q) by A1,A2,A3,A4,WEIERSTR:30;
A11: the distance of M is discerning by METRIC_1:def 7;
  now
    assume not min_dist_min(P,Q)>0;
    then dist(x1,x2)=0 by A1,A2,A3,A4,A10,Th36;
    then (the distance of M).(x1,x2)=0 by METRIC_1:def 1;
    then x1=x2 by A11,METRIC_1:def 3;
    hence P /\ Q <> {} by A9,XBOOLE_0:def 4;
  end;
  hence thesis by A5;
end;
