reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem
  X <> {} & X c= dom R implies R.:X <> {}
proof
  assume that
A1: X <> {} and
A2: X c= dom R;
  set x = the Element of X;
A3: x in dom R by A1,A2;
  then ex y being object st [x,y] in R by XTUPLE_0:def 12;
  hence thesis by A1,A3,Th104;
end;
