theorem
  H is existential implies bound_in H = bound_in the_argument_of H &
  the_scope_of H = the_argument_of the_scope_of the_argument_of H
proof
  assume H is existential;
  then H = Ex(bound_in H, the_scope_of H) by ZF_LANG:45;
  then
A1: the_argument_of H = All(bound_in H, 'not' the_scope_of H) by Th3;
  hence bound_in H = bound_in the_argument_of H by Th8;
  'not' the_scope_of H = the_scope_of the_argument_of H by A1,Th8;
  hence thesis by Th3;
end;
