theorem Th9:
  (for d holds a is_a_value_on d) & (for d holds b is_a_value_on d) implies
  rng <:denaming(V,A,a), denaming(V,A,b):> c= [:A,A:]
  proof
    set Da = denaming(V,A,a);
    set Db = denaming(V,A,b);
A1: rng <:Da,Db:> c= [:rng Da,rng Db:] by FUNCT_3:51;
    assume (for d holds a is_a_value_on d) & (for d holds b is_a_value_on d);
    then rng Da c= A & rng Db c= A by Th8;
    then [:rng Da,rng Db:] c= [:A,A:] by ZFMISC_1:96;
    hence thesis by A1,XBOOLE_1:1;
  end;
