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;

theorem
  for t1,t2 st t2 = (canonical_homomorphism T).t1 holds
  (canonical_homomorphism T).t1 = (canonical_homomorphism T).t2
  proof set H = canonical_homomorphism T;
    let t1,t2;
    assume Z0: t2 = H.t1;
    reconsider t = t1 as Element of (the Sorts of Free(S,X)).the_sort_of t1
    by SORT;
A1: the_sort_of @(H.t1) = the_sort_of (H.t1) &
    H.the_sort_of t1 is Function of (the Sorts of Free(S,X)).the_sort_of t1,
    (the Sorts of T).the_sort_of t1 by Lem00;
A2: dom(H**H) = (dom H)/\dom H = dom H = the carrier of S
    by PARTFUN1:def 2,PBOOLE:def 19;
    H**H = H by MSAFREE4:48;
    hence H.t1 = (H**H).(the_sort_of t1).t1 by ABBR
    .= ((H.the_sort_of t1)*(H.the_sort_of t1)).t by A2,PBOOLE:def 19
    .= (H.the_sort_of t1).((H.the_sort_of t1).t1) by FUNCT_2:15
    .= (H.the_sort_of t1).(H.t1) by ABBR
    .= (H.the_sort_of t2).(H.t1) by Z0,A1,Lem0
    .= H.t2 by Z0,ABBR;
  end;
