
theorem Th17:
  for M being non empty MetrSpace, A being Subset of M, B being
non empty Subset of M, C being Subset of M|B st A = C & C is bounded holds A is
  bounded
proof
  let M be non empty MetrSpace, A be Subset of M, B be non empty Subset of M,
  C be Subset of M|B;
  assume that
A1: A = C and
A2: C is bounded;
  consider r0 being Real such that
A3: 0 < r0 and
A4: for x, y being Point of M|B st x in C & y in C holds dist(x,y) <= r0
  by A2,TBSP_1:def 7;
  for x, y being Point of M st x in A & y in A holds dist(x,y) <= r0
  proof
    let x, y be Point of M;
    assume
A5: x in A & y in A;
    then reconsider x0 = x, y0 = y as Point of M|B by A1;
A6: (the distance of (M|B)).(x0,y0) = (the distance of M).(x,y) & (the
    distance of (M|B)).(x0,y0) = dist(x0,y0) by METRIC_1:def 1,TOPMETR:def 1;
    dist(x0,y0) <= r0 by A1,A4,A5;
    hence thesis by A6,METRIC_1:def 1;
  end;
  hence thesis by A3,TBSP_1:def 7;
end;
