reserve A,B,X,X1,Y,Y1,Y2,Z for set, a,x,y,z for object;
reserve P,R for Relation of X,Y;
reserve D,D1,D2,E,F for non empty set;
reserve R for Relation of D,E;
reserve x for Element of D;
reserve y for Element of E;

theorem
  dom R <> {} implies ex y being Element of E st y in rng R
proof
  assume dom R <> {};
  then rng R <> {} by RELAT_1:42;
  then ex y being object st y in rng R by XBOOLE_0:def 1;
  hence thesis;
end;
