
theorem Th34:
  for L being Abelian add-associative right_zeroed
  right_complementable well-unital associative commutative distributive
almost_left_invertible non empty doubleLoopStr for p,s being Polynomial of L
  st s <> 0_.(L) holds s divides p iff ex t being Polynomial of L st t *' s = p
proof
  let L be Abelian add-associative right_zeroed right_complementable
  well-unital associative commutative distributive almost_left_invertible non
  empty doubleLoopStr;
  let p,s be Polynomial of L;
  assume
A1: s <> 0_.(L);
A2: now
    deg(s) - 0 > 0 - 1 by A1,Lm8;
    then
A3: deg 0_.(L) < deg s by Th20;
    given t being Polynomial of L such that
A4: t *' s = p;
    p = t *' s + 0_.(L) by A4,POLYNOM3:28;
    then p div s = t by A3,Def5;
    then p mod s = 0_.(L) by A4,POLYNOM3:29;
    hence s divides p;
  end;
  now
    assume
A5: s divides p;
    consider t being Polynomial of L such that
A6: p = (p div s) *' s + t and
    deg t < deg s by A1,Def5;
    p mod s = t + ((p div s) *' s - ((p div s) *' s)) by A6,POLYNOM3:26
      .= t + 0_.(L) by POLYNOM3:29
      .= t by POLYNOM3:28;
    then t = 0_.(L) by A5;
    then p = (p div s) *' s by A6,POLYNOM3:28;
    hence ex t being Polynomial of L st t *' s = p;
  end;
  hence thesis by A2;
end;
