
theorem Th17: :: from POLYNOM4:5 right-distributive
  for L be add-associative right_zeroed right_complementable
  right-distributive non empty doubleLoopStr, p be Polynomial of L holds p*'(
  0_.(L)) = 0_.(L)
proof
  let L be add-associative right_zeroed right_complementable
  right-distributive non empty doubleLoopStr, p be Polynomial of L;
  now
    let i be Element of NAT;
    consider r be FinSequence of L such that
    len r = i+1 and
A1: (p*'(0_.(L))).i = Sum r and
A2: for k be Element of NAT st k in dom r holds r.k = p.(k-'1) * (0_.
    L).(i+1-'k) by POLYNOM3:def 9;
    now
      let k be Element of NAT;
      assume k in dom r;
      hence r.k = p.(k-'1) * (0_. L).(i+1-'k) by A2
        .= p.(k-'1) * 0.L by FUNCOP_1:7
        .= 0.L;
    end;
    hence (p*'(0_.(L))).i = 0.L by A1,POLYNOM3:1
      .= (0_.(L)).i by FUNCOP_1:7;
  end;
  hence thesis by FUNCT_2:63;
end;
