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
  X misses dom f implies f is_bounded_on X
proof
  assume X /\ dom f = {};
  then for c holds c in X /\ dom f implies ||.f/.c.|| <= 0;
  hence thesis;
end;
