reserve C for Simple_closed_curve,
  p,q,p1 for Point of TOP-REAL 2,
  i,j,k,n for Nat,
  r,s for Real;

theorem Th7:

:: JORDAN1K:39, JGRAPH_1:55
  for A,B being non empty compact Subset of TOP-REAL n st A misses B
  holds dist_min(A,B) > 0
proof
  let A,B be non empty compact Subset of TOP-REAL n such that
A1: A misses B;
A2: the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8;
  consider A9,B9 being Subset of TopSpaceMetr Euclid n such that
A3: A = A9 and
A4: B = B9 and
A5: dist_min(A,B) = min_dist_min(A9,B9) by JORDAN1K:def 1;
A6: A9 is compact by A2,A3,COMPTS_1:23;
  B9 is compact by A2,A4,COMPTS_1:23;
  hence thesis by A1,A3,A4,A5,A6,JGRAPH_1:38;
end;
