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
  for D being non empty set for F being ManySortedFunction of D for C
  being with_common_domain functional non empty set st C = rng F for d being
  Element of D,e being set st e in DOM C holds F.d.e = (commute F).e.d
proof
  let D be non empty set;
  let F be ManySortedFunction of D;
  set E = union the set of all  rng(F.d9) where d9 is Element of D ;
  reconsider F9= F as Function;
  let C be with_common_domain functional non empty set such that
A1: C = rng F;
A2: rng F9 c= Funcs(DOM C,E)
  proof
    let x be object;
    assume x in rng F9;
    then consider d9 be object such that
A3: d9 in dom F and
A4: F.d9 = x by FUNCT_1:def 3;
    reconsider d9 as Element of D by A3;
    consider Fd be Function such that
A5: Fd = F.d9;
A6: rng Fd c= E
    proof
A7:   rng Fd in the set of all  rng(F.d99) where d99 is Element of D  by A5;
      let x1 be object;
      assume x1 in rng Fd;
      hence thesis by A7,TARSKI:def 4;
    end;
    F.d9 in rng F by A3,FUNCT_1:def 3;
    then dom Fd = DOM C by A1,A5,CARD_3:108;
    hence thesis by A4,A5,A6,FUNCT_2:def 2;
  end;
  let d be Element of D,e be set;
  assume
A8: e in DOM C;
  dom F9 = D by PARTFUN1:def 2;
  then F in Funcs(D,Funcs(DOM C,E)) by A2,FUNCT_2:def 2;
  hence thesis by A8,FUNCT_6:56;
end;
