reserve x,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve V for RealNormSpace;
reserve f,f1,f2,f3 for PartFunc of C,V;
reserve r,r1,r2,p for Real;

theorem
  0(#)f is_bounded_on Y
proof
  now
    take p=0;
    let c;
    0*||.f/.c.|| = 0;
    then |.0 .|*||.f/.c.|| <= 0 by ABSVALUE:2;
    then
A1: ||.0*(f/.c).|| <= 0 by NORMSP_1:def 1;
    assume c in Y /\ dom (0(#)f);
    then c in dom (0(#)f) by XBOOLE_0:def 4;
    hence ||.(0(#)f)/.c.|| <= p by A1,Def4;
  end;
  hence thesis;
end;
