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 Th28:
  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 x be set st x in (the Sorts of U1).s holds
  (commute ((commute F).s)).x in product (Carrier(A,s))
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;
  (commute F).s in Funcs (I,Funcs(SU.s,SA)) by A1,Th26;
  then commute ((commute F).s) in Funcs (SU.s,Funcs(I,SA)) by FUNCT_6:55;
  then consider f9 be Function such that
A2: f9 = (commute (commute F).s) and
A3: dom f9 = SU.s and
A4: rng f9 c= Funcs(I,SA) by FUNCT_2:def 2;
  let x be set such that
A5: x in (the Sorts of U1).s;
  f9.x in rng f9 by A5,A3,FUNCT_1:def 3;
  then consider f be Function such that
A6: f = (commute (commute F).s).x and
A7: dom f = I and
  rng f c= SA by A2,A4,FUNCT_2:def 2;
A8: now
    let i1 be object;
    assume i1 in dom ((Carrier(A,s)));
    then reconsider i9 = i1 as Element of I;
    consider F1 be ManySortedFunction of U1,A.i9 such that
A9: F1 = F.i9 and
    F1 is_homomorphism U1,A.i9 by A1;
    x in dom (F1.s) by A5,FUNCT_2:def 1;
    then
A10: (ex U0 being MSAlgebra over S st U0 = A.i9 & (Carrier(A,s )).i9 = (
    the Sorts of U0).s )& F1.s.x in rng (F1.s) by FUNCT_1:def 3,PRALG_2:def 9;
    f.i9 = F1.s.x by A1,A5,A6,A9,Th27;
    hence ((commute ((commute F).s)).x).i1 in ((Carrier(A,s))).i1 by A6,A10;
  end;
  dom ((commute (commute F).s).x) = dom (Carrier(A,s)) by A6,A7,PARTFUN1:def 2;
  hence thesis by A8,CARD_3:9;
end;
