theorem Th42:
  for w being Element of Args(o,Free(S,Z)) for t being Element of Free(S,Z)
  st w is z-context_including &
  the_sort_of t = (the_arity_of o).(z-context_pos_in w)
  holds w+*(z-context_pos_in w,t) in Args(o,Free(S,Z))
  proof
    let w be Element of Args(o,Free(S,Z));
    let t be Element of Free(S,Z);
    assume
A1: w is z-context_including &
    the_sort_of t = (the_arity_of o).(z-context_pos_in w);
A2: dom (w+*(z-context_pos_in w,t)) = dom w = dom the_arity_of o
    by FUNCT_7:30,MSUALG_6:2;
    then
A3: len (w+*(z-context_pos_in w,t)) = len the_arity_of o by FINSEQ_3:29;
    now
      let i; assume
A4:   i in dom w;
      per cases;
      suppose i = z-context_pos_in w;
        then (w+*(z-context_pos_in w,t)).i = t &
        the_sort_of t = (the_arity_of o)/.i
        by A1,A2,A4,PARTFUN1:def 6,FUNCT_7:31;
        hence (w+*(z-context_pos_in w,t)).i in
        (the Sorts of Free(S,Z)).((the_arity_of o)/.i) by SORT;
      end;
      suppose i <> z-context_pos_in w;
        then (w+*(z-context_pos_in w,t)).i = w.i by FUNCT_7:32;
        hence (w+*(z-context_pos_in w,t)).i in
        (the Sorts of Free(S,Z)).((the_arity_of o)/.i) by A2,A4,MSUALG_6:2;
      end;
    end;
    hence w+*(z-context_pos_in w,t) in Args(o,Free(S,Z)) by A2,A3,MSAFREE2:5;
  end;
