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 Th157:
  Hom(T,x1,x2)**Hom(T,x1,x2) = id the Sorts of T
  proof
    set h = Hom(T,x1,x2);
    for s,x holds (h**h).s.(x-term) = x-term
    proof
      let s1,x11;
A2:   (FreeGen T).s1 c= (the Sorts of T).s1 by PBOOLE:def 2,def 18;
A3:   x11-term in FreeGen(s1,X) = (FreeGen T).s1 by MSAFREE:def 15,def 16;
A4:   ((h.s1)*(h.s1)).(x11-term) = (h.s1).((h.s1).(x11-term))
      by A2,A3,FUNCT_2:15;
      per cases;
      suppose s1 = s & x11 = x1;
        then h.s1.(x11-term) = x2-term & h.s1.(x2-term) = x11-term by HOM;
        hence thesis by A4,MSUALG_3:2;
      end;
      suppose s1 = s & x11 = x2;
        then h.s1.(x11-term) = x1-term & h.s1.(x1-term) = x11-term by HOM;
        hence thesis by A4,MSUALG_3:2;
      end;
      suppose s1 <> s or x11 <> x1 & x11 <> x2;
        then h.s1.(x11-term) = x11-term by HOM;
        hence thesis by A4,MSUALG_3:2;
      end;
    end;
    hence thesis by Th52;
  end;
