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;
