theorem Th8:
  (for d holds a is_a_value_on d) implies rng denaming(V,A,a) c= A
  proof
    assume
A1: for d holds a is_a_value_on d;
    set f = denaming(V,A,a);
    let y be object;
    assume y in rng f;
    then consider x being object such that
A2: x in dom f and
A3: f.x = y by FUNCT_1:def 3;
    dom f = {d where d is NonatomicND of V,A: a in dom d} by NOMIN_1:def 18;
    then consider d being NonatomicND of V,A such that
A4: x = d and a in dom d by A2;
    a is_a_value_on d by A1;
    hence thesis by A3,A4;
  end;
