
theorem lemNor1dega:
for F being Field
for G being non empty FinSequence of the carrier of Polynom-Ring F
holds Product G = 0_.(F) iff ex i being Element of dom G st G.i = 0_.(F)
proof
let F be Field, G be non empty FinSequence of the carrier of Polynom-Ring F;
A: now assume ex i being Element of dom G st G.i = 0_.(F); then
   consider i being Element of dom G such that B: G.i = 0_.(F);
   reconsider p = Product(<*G.i*>) as Polynomial of F by POLYNOM3:def 10;
   dom G = Seg(len G) by FINSEQ_1:def 3; then
   1 <= i & i <= len G by FINSEQ_1:1; then
   G = (G|(i-'1)) ^ <*G.i*> ^ (G/^i) by FINSEQ_5:84; then
   D: Product G
       = Product(G|(i-'1)^<*G.i*>) * Product((G/^i)) by GROUP_4:5
      .= (Product(G|(i-'1)) * Product(<*G.i*>)) * Product(G/^i) by GROUP_4:5
      .= (Product(<*G.i*>) * Product(G|(i-'1))) * Product(G/^i)
         by GROUP_1:def 12
      .= Product(<*G.i*>) *(Product(G|(i-'1)) * Product(G/^i))
         by GROUP_1:def 3;
    reconsider r1 = Product(G|(i-'1)),
               r2 = Product(G/^i) as Polynomial of F by POLYNOM3:def 10;
    F: p = 0_.(F) by B,GROUP_4:9;
    r1 *' r2 = Product(G|(i-'1)) * Product(G/^i) by POLYNOM3:def 10;
    hence Product G  = (0_.F) *' (r1 *' r2) by F,D,POLYNOM3:def 10
                    .= 0_.F;
    end;
defpred P[Nat] means
  for G being non empty FinSequence of the carrier of Polynom-Ring F
  st len G = $1 & for i being Element of dom G holds G.i <> 0_.(F)
  holds Product G <> 0_.(F);
C: P[1]
   proof
   now let G be non empty FinSequence of the carrier of Polynom-Ring F;
   assume C1: len G = 1 & for i being Element of dom G holds G.i <> 0_.(F);
   then C2: G = <*G.1*> by FINSEQ_1:40;
   then C3: dom G = Seg 1 by FINSEQ_1:38;
   then C4: 1 in dom G by FINSEQ_1:3;
   G = <*G/.1*> by C2,C3,PARTFUN1:def 6,FINSEQ_1:3;
   then Product G = G/.1 by GROUP_4:9 .= G.1 by C3,PARTFUN1:def 6,FINSEQ_1:3;
   hence Product G <> 0_.(F) by C1,C4;
   end;
   hence thesis;
   end;
D: now let k be Nat;
   assume 1 <= k;
   assume D2: P[k];
   now let G be non empty FinSequence of the carrier of Polynom-Ring F;
     assume D3: len G = k + 1 &
                for i being Element of dom G holds G.i <> 0_.(F);
     consider G1 being FinSequence, y being object such that
     B2: G = G1^<*y*> by FINSEQ_1:46;
     H: rng G c= the carrier of Polynom-Ring F by FINSEQ_1:def 4;

        rng G1 c= rng G by B2,FINSEQ_1:29; then
     reconsider G1 as FinSequence of the carrier of Polynom-Ring F
                                         by XBOOLE_1:1,H,FINSEQ_1:def 4;
     len <*y*> = 1 by FINSEQ_1:40; then
     C4: dom G = Seg(len G1 + 1) by B2,FINSEQ_1:def 7;
     B12: 1 <= len G1 + 1 by NAT_1:11; then
     B21: len G1 + 1 in dom G by C4,FINSEQ_1:1;
     dom <*y*> = Seg 1 by FINSEQ_1:38; then
     B11: 1 in dom <*y*> by FINSEQ_1:3;
     B20: y = <*y*>.1
           .= G.(len G1 + 1) by B11,B2,FINSEQ_1:def 7; then
     B13: y in rng G by B12,C4,FINSEQ_1:1,FUNCT_1:3;
     rng G c= the carrier of Polynom-Ring F by FINSEQ_1:def 4; then
     reconsider y as Element of the carrier of Polynom-Ring F by B13;
     B3: Product G = (Product G1) * y by B2,GROUP_4:6;
     B4: len G = len G1 + len<*y*> by B2,FINSEQ_1:22
              .= len G1 + 1 by FINSEQ_1:39;
     now assume Product G = 0_.(F); then
       Product G = 0.(Polynom-Ring F) by POLYNOM3:def 10; then
       per cases by B3,VECTSP_2:def 1;
       suppose y = 0.(Polynom-Ring F);
         then y = 0_.(F) by POLYNOM3:def 10;
         hence contradiction by D3,B20,B21;
         end;
       suppose D6: Product G1 = 0.(Polynom-Ring F);
         D7: Product(<*>(the carrier of Polynom-Ring F)) = 1_(Polynom-Ring F)
             by GROUP_4:8; then
         D9: G1 is non empty by D6; then
         D8: dom G1 <> {};
         Product G1 = 0_.(F) by D6,POLYNOM3:def 10; then
         consider i being Element of dom G1 such that
         D4: G1.i = 0_.(F) by D3,D2,B4,D7;
         dom G1 c= dom G by B2,FINSEQ_1:26; then
         D5: i in dom G & i in dom G1 by D8;
         G1.i = G.i by D9,B2,FINSEQ_1:def 7;
         hence contradiction by D3,D4,D5;
         end;
       end;
     hence Product G <> 0_.(F);
     end;
   hence P[k+1];
   end;
I: for k being Nat st k >= 1 holds P[k] from NAT_1:sch 8(C,D);
now assume B: Product G = 0_.(F);
  consider n being Nat such that H: n = len G;
  n >= 1 by H,NAT_1:14;
  hence ex i being Element of dom G st G.i = 0_.(F) by B,I,H;
  end;
hence thesis by A;
end;
