reserve i,n,m for Nat,
  x,y,X,Y for set,
  r,s for Real;

theorem Th15:
  for M be non empty MetrSpace,A be non empty Subset of M for B be
Subset of M, B9 be Subset of M|A st B = B9 holds B9 is bounded iff B is bounded
proof
  let M be non empty MetrSpace,A be non empty Subset of M;
  let B be Subset of M, B9 be Subset of M|A such that
A1: B = B9;
  thus B9 is bounded implies B is bounded by A1,HAUSDORF:17;
  assume
A2: B is bounded;
  per cases;
  suppose
    B9={}(M|A);
    hence thesis;
  end;
  suppose
    B9<>{}(M|A);
    then consider p be object such that
A3: p in B9 by XBOOLE_0:def 1;
    reconsider p as Point of (M|A) by A3;
A4: now
      let q be Point of (M|A) such that
A5:   q in B9;
      reconsider p9=p,q9=q as Point of M by TOPMETR:8;
A6:   dist(p,q) = dist(p9,q9) by TOPMETR:def 1;
A7:   diameter B+0<=diameter B+1 by XREAL_1:8;
      dist(p9,q9)<=diameter B by A1,A2,A3,A5,TBSP_1:def 8;
      hence dist(p,q)<=diameter B+1 by A6,A7,XXREAL_0:2;
    end;
    0+0 < diameter B+1 by A2,TBSP_1:21,XREAL_1:8;
    hence thesis by A4,TBSP_1:10;
  end;
end;
