
theorem
  for M being non empty MetrSpace, B being Subset of M, A being Subset
  of TopSpaceMetr M st A = B & A is compact holds B is bounded
proof
  let M be non empty MetrSpace, B be Subset of M, A be Subset of TopSpaceMetr
  M;
  set TA = TopSpaceMetr M;
  assume that
A1: A = B and
A2: A is compact;
  A c= the carrier of (TA|A) by PRE_TOPC:8;
  then reconsider A2 = A as Subset of (TA|A);
  per cases;
  suppose
    A <> {};
    then reconsider A1 = A as non empty Subset of M by TOPMETR:12;
    [#](TA|A) = A2 by PRE_TOPC:def 5;
    then [#](TA|A) is compact by A2,COMPTS_1:2;
    then
A3: TA|A is compact by COMPTS_1:1;
    TopSpaceMetr (M|A1) = TA|A by Th16;
    then M|A1 is totally_bounded by A3,TBSP_1:9;
    then M|A1 is bounded by TBSP_1:19;
    then
A4: [#](M|A1) is bounded;
    [#](M|A1) = the carrier of M|A1 .= A1 by TOPMETR:def 2;
    hence thesis by A1,A4,Th17;
  end;
  suppose
    A = {};
    then A = {} M;
    hence thesis by A1;
  end;
end;
