theorem
  Y <> {} & Y c= rng R implies R"Y <> {}
proof
  assume that
A1: Y <> {} and
A2: Y c= rng R;
  set y = the Element of Y;
A3: y in rng R by A1,A2;
  then ex x being object st [x,y] in R by XTUPLE_0:def 13;
  hence thesis by A1,A3,Th123;
end;
