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
  Y c= X & f is_bounded_on X implies f is_bounded_on Y
proof
  assume that
A1: Y c= X and
A2: f is_bounded_on X;
  consider r such that
A3: for c st c in X /\ dom f holds ||.f/.c.|| <= r by A2;
  take r;
  let c;
  assume c in Y /\ dom f;
  then c in Y & c in dom f by XBOOLE_0:def 4;
  then c in X /\ dom f by A1,XBOOLE_0:def 4;
  hence thesis by A3;
end;
