reserve I for non empty set,
  J for ManySortedSet of I,
  S for non void non empty ManySortedSign,
  i for Element of I,
  c for set,
  A for MSAlgebra-Family of I,S,
  EqR for Equivalence_Relation of I,
  U0,U1,U2 for MSAlgebra over S,
  s for SortSymbol of S,
  o for OperSymbol of S,
  f for Function;

theorem Th27:
  for U1 be non-empty MSAlgebra over S
  for F being ManySortedFunction of I st
  (for i be Element of I holds ex F1 being ManySortedFunction of U1,A.i st
  F1 = F.i & F1 is_homomorphism U1,A.i)
  for F9 be ManySortedFunction of U1,A.i st F9 = F.i
  for x be set st x in (the Sorts of U1).s
  for f be Function st f = (commute((commute F).s)).x holds f.i = F9.s.x
proof
  let U1 be non-empty MSAlgebra over S;
  set SU = the Sorts of U1, SA = union the set of all
(the Sorts of A.i9).s1 where i9 is
  Element of I,s1 is SortSymbol of S ;
  let F be ManySortedFunction of I such that
A1: for i be Element of I holds ex F1 being ManySortedFunction of U1,A.i
  st F1 = F.i & F1 is_homomorphism U1,A.i;
A2: (commute F).s in Funcs (I,Funcs(SU.s,SA)) by A1,Th26;
  then dom ((commute F).s) = I by FUNCT_2:92;
  then
A3: ((commute F).s).i in rng ((commute F).s) by FUNCT_1:def 3;
  reconsider f9 = (commute F).s as Function;
  rng ((commute F).s) c= Funcs(SU.s,SA) by A2,FUNCT_2:92;
  then consider g be Function such that
A4: g = f9.i and
  dom g = SU.s and
  rng g c= SA by A3,FUNCT_2:def 2;
  let F9 be ManySortedFunction of U1,A.i such that
A5: F9 = F.i;
  let x1 be set such that
A6: x1 in (the Sorts of U1).s;
  let f be Function such that
A7: f = (commute((commute F).s)).x1;
  g = F9.s by A1,A5,A4,Th25;
  hence thesis by A6,A7,A2,A4,FUNCT_6:56;
end;
