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;
