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);
reserve h1 for x-constant Homomorphism of Free(S,X), T,
  h2 for y-constant Homomorphism of Free(S,Y), Q;
reserve
  s2 for s1-reachable SortSymbol of S,
  g1 for Translation of Free(S,Y),s1,s2,
  g for Translation of Free(S,X),s1,s2;

theorem Th91:
  for s1,s2 being SortSymbol of S, V being finite set holds
  m in dom q & s1 = (the_arity_of o)/.m implies
  ex y being (Element of Y.s1), C being (context of y), q1 st
  y nin V & q1 = q+*(m,y-term) &
  q1 is y-context_including & y-context_in q1 = y-term & C = o-term q1 &
  m = y-context_pos_in q1 & transl C = transl(o,m,q,Free(S,Y))
  proof
    let s1,s2 be SortSymbol of S, V be finite set;
    assume Z1: m in dom q;
    assume Z2: s1 = (the_arity_of o)/.m;
    consider y being (Element of Y.s1), C being (context of y), q1 such that
A1: y nin V & C = o-term q1 & q1 = q+*(m,y-term) &
    q1 is y-context_including &
    m = y-context_pos_in q1 & y-context_in q1 = y-term by Z1,Z2,Th59;
    take y, C, q1;
    thus y nin V by A1;
    thus q1 = q+*(m,y-term) by A1;
    thus q1 is y-context_including by A1;
    thus y-context_in q1 = y-term by A1;
    thus C = o-term q1 by A1;
    thus m = y-context_pos_in q1 by A1;
    dom transl C = (the Sorts of Free(S,Y)).s1 by FUNCT_2:def 1;
    hence dom transl C = dom transl(o,m,q,Free(S,Y)) by Z2,MSUALG_6:def 4;
    let c be object; assume
    c in dom transl C;
    then reconsider c as Element of (the Sorts of Free(S,Y)).s1;
    reconsider q2 = q+*(m,c) as Element of Args(o,Free(S,Y))
    by Z2,MSUALG_6:7;
A6: transl(o,m,q,Free(S,Y)).c = Den(o,Free(S,Y)).q2
    by Z2,MSUALG_6:def 4
    .= o-term q2 by MSAFREE4:13;
    q2 = q1+*(m,c) by A1,FUNCT_7:34;
    then C-sub c = o-term q2 & the_sort_of c = s1 by A1,Th58,SORT;
    hence thesis by A6,TRANS;
  end;
