reserve r,t for Real;
reserve i for Integer;
reserve k,n for Nat;
reserve p for Polynomial of F_Real;
reserve e for Element of F_Real;
reserve L for non empty ZeroStr;
reserve z,z0,z1,z2 for Element of L;

theorem Th17: :: POLYNOM3:34'
  for L being add-associative right_zeroed right_complementable
        left-distributive non empty doubleLoopStr
  for p being 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 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:def 8;
  end;
