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 Th26:
  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) holds
  (commute F).s in Funcs(I,Funcs((the Sorts of U1).s, union the set of all
  (the Sorts of A.i9).s1 where i9 is Element of I,s1 is SortSymbol of S ))
proof
  let U1 be non-empty MSAlgebra over 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;
  set SU = the Sorts of U1, CA = the carrier of S, SA = union the set of all
(the Sorts of A
  .i9).s1 where i9 is Element of I,s1 is SortSymbol of S;
set SA9 = the set of all
(the Sorts of A.i9).s1 where i9 is Element of I,s1 is SortSymbol of S;
set FS =
the set of all F.i9.s1 where s1 is SortSymbol of S,i9 is Element of
I;
  F in Funcs(I,Funcs(CA,FS)) by A1,Th25;
  then commute F in Funcs(CA,Funcs(I,FS)) by FUNCT_6:55;
  then consider F9 be Function such that
A2: F9 = commute F & dom F9 = CA and
A3: rng F9 c= Funcs(I,FS) by FUNCT_2:def 2;
  (commute F).s in rng F9 by A2,FUNCT_1:def 3;
  then
A4: ex F2 be Function st F2 = (commute F).s & dom F2 = I & rng F2 c= FS by A3,
FUNCT_2:def 2;
  rng ((commute F).s) c= Funcs(SU.s,SA)
  proof
    let x9 be object;
    assume x9 in rng ((commute F).s);
    then consider i9 be object such that
A5: i9 in dom ((commute F).s) and
A6: x9 = ((commute F).s).i9 by FUNCT_1:def 3;
    reconsider i1 = i9 as Element of I by A4,A5;
    consider F9 be ManySortedFunction of U1,A.i1 such that
A7: F9 = F.i1 and
    F9 is_homomorphism U1,A.i1 by A1;
    (the Sorts of A.i1).s c= SA
    proof
A8:   (the Sorts of A.i1).s in SA9;
      let y be object;
      assume y in (the Sorts of A.i1).s;
      hence thesis by A8,TARSKI:def 4;
    end;
    then
A9: dom (F9.s) = SU.s & rng (F9.s) c= SA by FUNCT_2:def 1;
    x9 = F9.s by A1,A6,A7,Th25;
    hence thesis by A9,FUNCT_2:def 2;
  end;
  hence thesis by A4,FUNCT_2:def 2;
end;
