reserve
  a,b for object, I,J for set, f for Function, R for Relation,
  i,j,n for Nat, m for (Element of NAT),
  S for non empty non void ManySortedSign,
  s,s1,s2 for SortSymbol of S,
  o for OperSymbol of S,
  X for non-empty ManySortedSet of the carrier of S,
  x,x1,x2 for (Element of X.s), x11 for (Element of X.s1),
  T for all_vars_including inheriting_operations free_in_itself
  (X,S)-terms MSAlgebra over S,
  g for Translation of Free(S,X),s1,s2,
  h for Endomorphism of Free(S,X);
reserve
  r,r1,r2 for (Element of T),
  t,t1,t2 for (Element of Free(S,X));
reserve
  Y for infinite-yielding ManySortedSet of the carrier of S,
  y,y1 for (Element of Y.s), y11 for (Element of Y.s1),
  Q for all_vars_including inheriting_operations free_in_itself
  (Y,S)-terms MSAlgebra over S,
  q,q1 for (Element of Args(o,Free(S,Y))),
  u,u1,u2 for (Element of Q),
  v,v1,v2 for (Element of Free(S,Y)),
  Z for non-trivial ManySortedSet of the carrier of S,
  z,z1 for (Element of Z.s),
  l,l1 for (Element of Free(S,Z)),
  R for all_vars_including inheriting_operations free_in_itself
  (Z,S)-terms MSAlgebra over S,
  k,k1 for Element of Args(o,Free(S,Z));
reserve c,c1,c2 for set, d,d1 for DecoratedTree;
reserve
  w for (Element of Args(o,T)),
  p,p1 for Element of Args(o,Free(S,X));
reserve C for (context of x), C1 for (context of y), C9 for (context of z),
  C11 for (context of x11), C12 for (context of y11), D for context of s,X;
reserve
  S9 for sufficiently_rich non empty non void ManySortedSign,
  s9 for SortSymbol of S9,
  o9 for s9-dependent OperSymbol of S9,
  X9 for non-trivial ManySortedSet of the carrier of S9,
  x9 for (Element of X9.s9);

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;
