
theorem
  for R being commutative non empty multMagma, a being Element of R,
  p being FinSequence of the carrier of R holds p * a = a * p
proof
  let R be commutative non empty multMagma, a be Element of R, p be
  FinSequence of the carrier of R;
  set pa = p*a, ap = a*p;
A1: dom pa = dom p by POLYNOM1:def 2;
A2: dom ap = dom p by POLYNOM1:def 1;
  now
    let i be Nat such that
A3: i in dom pa;
    thus pa/.i = p/.i * a by A1,A3,POLYNOM1:def 2
    .= ap/.i by A1,A3,POLYNOM1:def 1;
  end;
  hence thesis by A1,A2,FINSEQ_5:12;
end;
