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 Th7:
  the_arity_of o = {} implies (commute (OPER A)).o in Funcs(I,Funcs
  ({{}}, union the set of all  Result(o,A.i9) where i9 is Element of I))
proof
  set f = (commute (OPER A)).o;
  set C = union the set of all  Result(o,A.i9) where i9 is Element of I;
  commute (OPER A) in Funcs(the carrier' of S, Funcs(I,rng uncurry (OPER A
  ))) by PRALG_2:6;
  then
A1: ex f1 be Function st commute (OPER A) = f1 & dom f1 = the carrier' of S &
  rng f1 c= Funcs(I,rng uncurry (OPER A)) by FUNCT_2:def 2;
  then f in rng commute (OPER A) by FUNCT_1:def 3;
  then
A2: ex fb be Function st fb = f & dom fb = I & rng fb c= rng uncurry (OPER A)
  by A1,FUNCT_2:def 2;
  assume
A3: the_arity_of o = {};
  now
    let x be object;
    assume x in rng f;
    then consider a be object such that
A4: a in dom f and
A5: f.a = x by FUNCT_1:def 3;
    f = A?.o;
    then reconsider x9 = x as Function by A5;
    reconsider a as Element of I by A2,A4;
A6: x9 = (A?.o).a by A5
      .= Den(o,A.a) by PRALG_2:7;
    then
A7: dom x9 = Args(o,A.a) by FUNCT_2:def 1
      .= {{}} by A3,PRALG_2:4;
    now
      let c be object;
      assume c in rng x9;
      then consider b be object such that
A8:   b in dom x9 and
A9:   x9.b = c by FUNCT_1:def 3;
      x9.b = const(o,A.a) by A6,A7,A8,TARSKI:def 1;
      then
A10:  c is Element of Result(o,A.a) by A3,A9,Th5;
      Result(o,A.a) in the set of all  Result(o,A.i9) where i9 is Element of I;
      hence c in C by A10,TARSKI:def 4;
    end;
    then rng x9 c= C;
    hence x in Funcs({{}},C) by A7,FUNCT_2:def 2;
  end;
  then rng f c= Funcs({{}},C);
  hence thesis by A2,FUNCT_2:def 2;
end;
