reserve n for Element of NAT;

theorem
  for P, Q, R being non empty Subset of TOP-REAL n st P is compact & Q
is compact & R is compact holds HausDist (P, R) <= HausDist (P, Q) + HausDist (
  Q, R)
proof
  let P, Q, R be non empty Subset of TOP-REAL n;
  assume that
A1: P is compact & Q is compact and
A2: R is compact;
A3: the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8;
  then reconsider
  P1 = P, Q1 = Q, R1 = R as non empty Subset of TopSpaceMetr Euclid
  n;
A4: R1 is compact by A2,A3,COMPTS_1:23;
A5: HausDist (Q1, R1) = HausDist (Q, R) by Def3;
A6: HausDist (P1, R1) = HausDist (P, R) & HausDist (P1, Q1) = HausDist (P, Q
  ) by Def3;
  P1 is compact & Q1 is compact by A1,A3,COMPTS_1:23;
  hence thesis by A4,A6,A5,Th38;
end;
