reserve n for Element of NAT;

theorem
  for P, Q being non empty Subset of TOP-REAL n st P is compact & Q is
  compact & HausDist (P, Q) = 0 holds P = Q
proof
  let P, Q be non empty Subset of TOP-REAL n;
  assume that
A1: P is compact & Q is compact and
A2: HausDist (P, Q) = 0;
A3: the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8;
  then reconsider P1 = P, Q1 = Q as non empty Subset of TopSpaceMetr Euclid n;
A4: HausDist (P1, Q1) = 0 by A2,Def3;
  P1 is compact & Q1 is compact by A1,A3,COMPTS_1:23;
  hence thesis by A4,Th37;
end;
