reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;

theorem Th1:
  dom (g^) c= dom g & dom g /\ (dom g \ g"{0}) = dom g \ g"{0}
proof
  dom (g^) = dom g \ g"{0} by Def2;
  hence dom (g^) c= dom g;
  thus thesis by XBOOLE_1:28;
end;
