
theorem Th16:
  for M being non empty MetrSpace, P being Subset of TopSpaceMetr
  M, Q being non empty Subset of M holds P = Q implies (TopSpaceMetr M)|P =
  TopSpaceMetr(M|Q)
proof
  let M be non empty MetrSpace, P be Subset of TopSpaceMetr M, Q be non empty
  Subset of M;
  reconsider N = TopSpaceMetr(M|Q) as SubSpace of TopSpaceMetr M by TOPMETR:13;
A1: the carrier of N = the carrier of M|Q by TOPMETR:12;
  assume P = Q;
  then [#]N = P by A1,TOPMETR:def 2;
  hence thesis by PRE_TOPC:def 5;
end;
