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;

theorem
  for r being Element of T,s holds
  Hom(T,x1,x2).s.r = ((canonical_homomorphism T)**Hom(Free(S,X),x1,x2)).s.r
  proof let r be Element of T,s;
    the Sorts of T is MSSubset of Free(S,X) by MSAFREE4:def 6;
    then
A1: (the Sorts of T).s c= (the Sorts of Free(S,X)).s
    by PBOOLE:def 2,def 18;
    ((canonical_homomorphism T)**Hom(Free(S,X),x1,x2)).s.r =
    (Hom(T,x1,x2)**canonical_homomorphism T).s.r =
    ((Hom(T,x1,x2).s)*((canonical_homomorphism T).s)).r =
    (Hom(T,x1,x2).s).((canonical_homomorphism T).s.r) =
    Hom(T,x1,x2).s.r by A1,Th66,MSUALG_3:2,FUNCT_2:15,MSAFREE4:47;
    hence thesis;
  end;
