reserve X for set,
  Y for non empty set;
reserve n for Nat;
reserve r for Real,
  M for non empty MetrSpace;
reserve n for Nat,
  p,q,q1,q2 for Point of TOP-REAL 2,
  r,s1,s2,t1,t2 for Real,
  x,y for Point of Euclid 2;

theorem Th39:
  for A,B being compact Subset of TOP-REAL n st A meets B holds
  dist_min(A,B) = 0
proof
  let A,B be compact Subset of TOP-REAL n such that
A1: A meets B;
  consider A9,B9 be Subset of TopSpaceMetr Euclid n such that
A2: A = A9 & B = B9 and
A3: dist_min(A,B) = min_dist_min(A9,B9) by Def1;
  the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8;
  then A9 is compact & B9 is compact by A2,COMPTS_1:23;
  hence thesis by A1,A2,A3,Th12;
end;
