
theorem Th2:
  for L be add-associative right_zeroed right_complementable
left-distributive non empty doubleLoopStr for p be sequence of L holds (0_.(L
  ))*'p = 0_.(L)
proof
  let L be add-associative right_zeroed right_complementable left-distributive
  non empty doubleLoopStr;
  let p be sequence of L;
  now
    let i be Element of NAT;
    consider r be FinSequence of the carrier of L such that
    len r = i+1 and
A1: ((0_.(L))*'p).i = Sum r and
A2: for k be Element of NAT st k in dom r holds r.k = (0_.(L)).(k-'1)
    * p.(i+1-'k) by POLYNOM3:def 9;
    now
      let k be Element of NAT;
      assume k in dom r;
      hence r.k = (0_.(L)).(k-'1) * p.(i+1-'k) by A2
        .= 0.L * p.(i+1-'k) by FUNCOP_1:7
        .= 0.L;
    end;
    hence ((0_.(L))*'p).i = 0.L by A1,POLYNOM3:1
      .= (0_.(L)).i by FUNCOP_1:7;
  end;
  hence thesis by FUNCT_2:63;
end;
