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 Th20:
  for I,S,A,i for o be OperSymbol of S st (the_arity_of o) <> {}
  for U1 be non-empty MSAlgebra over S for x be Element of Args(o,product A)
  holds (commute x).i is Element of Args(o,A.i)
proof
  let I,S,A,i;
  let o be OperSymbol of S such that
A1: (the_arity_of o) <> {};
  let U1 be non-empty MSAlgebra over S;
  let x be Element of Args(o,product A);
  i in I;
  then
A2: i in dom (doms(A?.o)) by PRALG_2:11;
  (commute x) in product doms(A?.o) by A1,Th17;
  then (commute x).i in doms(A?.o).i by A2,CARD_3:9;
  hence thesis by PRALG_2:11;
end;
