reserve
  r,s,r0,s0,t for Real;

theorem Th14:
  for S being Subset of TOP-REAL 2 st S is bounded holds
  proj2.:S is real-bounded
proof
  let S be Subset of TOP-REAL 2;
  assume S is bounded;
  then reconsider C = S as bounded Subset of Euclid 2 by JORDAN2C:11;
  consider r being Real, x being Point of Euclid 2 such that
A1: 0 < r and
A2: C c= Ball(x,r) by METRIC_6:def 3;
  reconsider P = Ball(x,r) as Subset of TOP-REAL 2 by TOPREAL3:8;
  reconsider p = x as Point of TOP-REAL 2 by TOPREAL3:8;
  set t = max(|.p`2-r.|,|.p`2+r.|);
  now
    assume that
A3: |.p`2-r.| <= 0 and
A4: |.p`2+r.| <= 0;
    |.p`2-r.| = 0 by A3,COMPLEX1:46;
    then |.r-p`2.| = 0 by UNIFORM1:11;
    then
A5: r-p`2 = 0 by ABSVALUE:2;
    |.p`2+r.| = 0 by A4,COMPLEX1:46;
    hence contradiction by A1,A5,ABSVALUE:2;
  end;
  then
A6: t > 0 by XXREAL_0:30;
A7: proj2.:P = ].p`2-r,p`2+r.[ by TOPREAL6:44;
  for s st s in proj2.:S holds |.s.| < t
  proof
    let s;
    proj2.:S c= proj2.:P by A2,RELAT_1:123;
    hence thesis by A7,Th3;
  end;
  hence thesis by A6,SEQ_4:4;
end;
