
theorem Th37:
  for M being non empty MetrSpace, P, Q being non empty Subset of
TopSpaceMetr M st P is compact & Q is compact & HausDist (P, Q) = 0 holds P = Q
proof
  let M be non empty MetrSpace, P, Q be non empty Subset of TopSpaceMetr M;
  assume that
A1: P is compact and
A2: Q is compact;
A3: Q is closed by A2,COMPTS_1:7;
  assume
A4: HausDist (P, Q) = 0;
  then max_dist_min (Q, P) = 0 by A1,A2,Th1,Th28;
  then upper_bound ((dist_min Q).:P) = 0 by WEIERSTR:def 8;
  then consider Y being non empty Subset of REAL such that
A5: (dist_min Q) .: P = Y and
A6: 0 = upper_bound Y by Th11;
A7: Y is bounded_above by A1,A2,A5,Th25;
  thus P c= Q
  proof
    let x be object;
    assume
A8: x in P;
    then reconsider x9 = x as Point of M by TOPMETR:12;
    dom dist_min Q = the carrier of TopSpaceMetr M by FUNCT_2:def 1;
    then (dist_min Q) . x in Y by A5,A8,FUNCT_1:def 6;
    then
A9: (dist_min Q) . x <= 0 by A6,A7,SEQ_4:def 1;
    (dist_min Q) . x >= 0 by A8,JORDAN1K:9;
    then (dist_min Q) . x = 0 by A9,XXREAL_0:1;
    then x9 in Q by A3,Th9;
    hence thesis;
  end;
  let x be object;
  assume
A10: x in Q;
  then reconsider x9 = x as Point of M by TOPMETR:12;
A11: P is closed by A1,COMPTS_1:7;
  max_dist_min (P, Q) = 0 by A1,A2,A4,Th1,Th28;
  then upper_bound ((dist_min P).:Q) = 0 by WEIERSTR:def 8;
  then consider X being non empty Subset of REAL such that
A12: (dist_min P) .: Q = X and
A13: 0 = upper_bound X by Th11;
  dom dist_min P = the carrier of TopSpaceMetr M by FUNCT_2:def 1;
  then
A14: (dist_min P) . x in X by A12,A10,FUNCT_1:def 6;
  X is bounded_above by A1,A2,A12,Th25;
  then
A15: (dist_min P) . x <= 0 by A13,A14,SEQ_4:def 1;
  (dist_min P) . x >= 0 by A10,JORDAN1K:9;
  then (dist_min P) . x = 0 by A15,XXREAL_0:1;
  then x9 in P by A11,Th9;
  hence thesis;
end;
