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 Th21:
  for I,S,A,i,o for x be Element of Args(o,product A) for n be set
st n in dom(the_arity_of o) for f be Function st f = x.n holds ((commute x).i).
  n = f.i
proof
  let I,S,A,i,o;
  let x be Element of Args(o,product A);
  let n be set such that
A1: n in dom the_arity_of o;
  set C = union the set of all (the Sorts of A.i9).s9 where
    i9 is Element of I,s9 is Element of (the carrier of S);
A2: x in Funcs (dom (the_arity_of o),Funcs (I,C)) by Th14;
  let g be Function;
  assume g = x.n;
  hence thesis by A1,A2,FUNCT_6:56;
end;
