reserve i,j,k for Nat;
reserve K for non empty addLoopStr,
  a for Element of K,
  p for FinSequence of the carrier of K,
  R for Element of i-tuples_on the carrier of K;
reserve K for left_zeroed right_zeroed add-associative right_complementable
  non empty addLoopStr,
  R,R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for non empty addLoopStr,
  a1,a2 for Element of K,
  p1,p2 for FinSequence of the carrier of K,
  R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for Abelian right_zeroed add-associative right_complementable non
  empty addLoopStr,
  R,R1,R2,R3 for Element of i-tuples_on the carrier of K;
reserve K for non empty multMagma,
  a,a9,a1,a2 for Element of K,
  p for FinSequence of the carrier of K,
  R for Element of i-tuples_on the carrier of K;
reserve K for distributive non empty doubleLoopStr,
  a,a1,a2 for Element of K ,
  R,R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for non empty multMagma,
  a1,a2,b1,b2 for Element of K,
  p1,p2 for FinSequence of the carrier of K,
  R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for commutative non empty multMagma,
  p,q for FinSequence of the carrier of K,
  R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for commutative associative non empty multMagma,
  a,a1,a2 for Element of K,
  R for Element of i-tuples_on the carrier of K;
reserve K for commutative associative non empty multMagma,
  a for Element of K,
  R,R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for add-associative right_zeroed right_complementable non empty
  addLoopStr,
  a for Element of K,
  p for FinSequence of the carrier of K;
reserve K for Abelian add-associative right_zeroed right_complementable non
  empty addLoopStr,
  p for FinSequence of the carrier of K,
  R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for commutative associative well-unital non empty doubleLoopStr,
  a ,a1,a2,a3 for Element of K,
  p1 for FinSequence of the carrier of K,
  R1,R2 for Element of i-tuples_on the carrier of K;

theorem
  for K being add-associative right_zeroed right_complementable Abelian
    commutative associative well-unital distributive almost_left_invertible
    non degenerated non empty doubleLoopStr
  for p being FinSequence of the carrier of K holds
    (ex k st k in dom p & p.k = 0.K) iff Product p = 0.K
proof
  let K be add-associative right_zeroed right_complementable Abelian
  commutative associative well-unital distributive almost_left_invertible non
  degenerated non empty doubleLoopStr;
  let p be FinSequence of the carrier of K;
  defpred P[Nat] means for p be FinSequence of the carrier of K st
  len p = $1 holds (ex k st k in Seg $1 & p.k = 0.K) iff Product p = 0.K;
A1: for i st P[i] holds P[i+1]
  proof
    let i such that
A2: for p be FinSequence of the carrier of K st len p = i
     holds (ex k st k in Seg i & p.k =0.K) iff Product p = 0.K;
    let p be FinSequence of the carrier of K;
    assume
A3: len p = i+1;
    then consider
    p9 be FinSequence of the carrier of K, a be Element of K such
    that
A4: p = p9^<*a*> by FINSEQ_2:19;
A5: i+ 1= len p9+ 1 by A3,A4,FINSEQ_2:16;
    then
A6: i=len p9 by XCMPLX_1:2;
A7: Product p = Product p9 * a by A4,GROUP_4:6;
    thus (ex k st k in Seg (i+1) & p.k = 0.K) implies Product p = 0.K
    proof
      given k such that
A8:   k in Seg (i+1) and
A9:   p.k = 0.K;
      now
        per cases by A8,FINSEQ_2:7;
        suppose
A10:      k in Seg i;
          then k in dom p9 by A6,FINSEQ_1:def 3;
          then p9.k = p.k by A4,FINSEQ_1:def 7;
          then Product p9 = 0.K by A2,A6,A9,A10;
          hence thesis by A7;
        end;
        suppose
          k = i+1;
          then a = 0.K by A4,A5,A9,FINSEQ_1:42;
          hence thesis by A7;
        end;
      end;
      hence thesis;
    end;
    assume
A11: Product p = 0.K;
    per cases by A7,A11,VECTSP_1:12;
    suppose
      Product p9 = 0.K;
      then consider k such that
A12:  k in Seg i and
A13:  p9.k = 0.K by A2,A6;
      k in dom p9 by A6,A12,FINSEQ_1:def 3;
      then p.k = 0.K by A4,A13,FINSEQ_1:def 7;
      hence thesis by A12,FINSEQ_2:8;
    end;
    suppose
      a = 0.K;
      then p.(i+1) = 0.K by A4,A5,FINSEQ_1:42;
      hence thesis by FINSEQ_1:4;
    end;
  end;
A14: Seg len p = dom p by FINSEQ_1:def 3;
A15: P[0]
  proof
    let p be FinSequence of the carrier of K;
    assume len p = 0;
    then p=<*>(the carrier of K);
    then
A16: Product p = 1.K by Lm6;
    thus (ex k st k in Seg 0 & p.k=0.K) implies Product p=0.K;
    assume Product p=0.K;
    hence thesis by A16;
  end;
  for i holds P[i] from NAT_1:sch 2(A15,A1);
  hence thesis by A14;
end;
