 reserve n,k for Nat;
 reserve L for comRing;
 reserve R for domRing;
 reserve x0 for positive Real;

theorem :::
  for a be Element of L, p be non empty FinSequence of the carrier of L
    st (for j be Nat st j in dom p holds a divides p/.j)
    holds a divides Sum p
    proof
      let a be Element of L, p be non empty FinSequence of the carrier of L;
      assume
A1:   for j be Nat st j in dom p holds a divides p/.j;
      for i being Nat st i in dom p holds p.i in {a}-Ideal
      proof
        let i be Nat;
        assume
A3:     i in dom p; then
        p.i = p/.i by PARTFUN1:def 6;
        hence thesis by A1,A3,RING_2:18;
      end;
      hence thesis by RING_2:18,Th13;
    end;
