
theorem DP1:
for p being Prime
for F being p-characteristic Field
for q being Element of the carrier of Polynom-Ring F
st for i being Nat st i in Support q
   holds p divides i & ex a being Element of F st a|^p = q.i
ex r being Element of the carrier of Polynom-Ring F st r|^p = q
proof
let p be Prime, F be p-characteristic Field;
let q be Element of the carrier of Polynom-Ring F;
assume AS: for i being Nat st i in Support q
           holds p divides i & ex a being Element of F st a|^p = q.i;
per cases;
suppose q is zero; then
  H: q = 0_.(F) by UPROOTS:def 5 .= 0.(Polynom-Ring F) by POLYNOM3:def 10;
  then q|^p = 0.(Polynom-Ring F);
  hence thesis by H;
  end;
suppose q is non zero; then
reconsider q as non zero Element of the carrier of Polynom-Ring F;
defpred P[Nat] means
 for q being non zero Element of the carrier of Polynom-Ring F
 st deg q = $1 * p & for i being Nat st i in Support q
    holds p divides i & ex a being Element of F st a|^p = q.i
 ex r being Element of the carrier of Polynom-Ring F st r|^p = q;
IS: now let k be Nat;
    assume IV: for n being Nat st n < k holds P[n];
    now let q be non zero Element of the carrier of Polynom-Ring F;
      assume A1: deg q = k * p & for i being Nat st i in Support q
         holds p divides i & ex a being Element of F st a|^p = q.i;
      H: q <> 0_.(F); then
      len q <> 0 by POLYNOM4:5; then
      K: len q -' 1 = len q - 1 by XREAL_0:def 2;
      consider r being Polynomial of F such that
      A2: len r < len q & q = r + LM q &
          for n be Element of NAT st n < len q-1 holds r.n = q.n
          by H,POLYNOM4:5,POLYNOM4:16;
      LC q is non zero; then
      J: q.(len q - 1) <> 0.F by K,RATFUNC1:def 6; then
      q.(deg q) <> 0.F by HURWITZ:def 2; then
      deg q in Support q by POLYNOM1:def 4; then
      consider b being Element of F such that
      A4: b|^p = q.(deg q) by A1;
      reconsider r1 = anpoly(b,k) as
            Element of the carrier of Polynom-Ring F by POLYNOM3:def 10;
      K: b is non zero by A4,J,HURWITZ:def 2;
      A5: LM q = anpoly(b|^p,k*p) by A4,A1,FIELD_1:11
            .= anpoly(b,k)`^p by K,DP3
            .= r1|^p;
      per cases;
      suppose r is zero;
        hence
           ex r being Element of the carrier of Polynom-Ring F st r|^p = q
           by A2,A5;
        end;
      suppose r is non zero; then
        reconsider r as non zero Element of the carrier of Polynom-Ring F
          by POLYNOM3:def 10;
        r <> 0_.(F); then
        len r <> 0 by POLYNOM4:5; then
        K: len r -' 1 = len r - 1 by XREAL_0:def 2;
        H: now let i be Element of NAT;
           assume i in Support r; then
           H1: r.i <> 0.F by POLYNOM1:def 4;
           now assume H2: i >= len q - 1;
               len q - 1 > len r - 1 by A2,XREAL_1:9;
               then i > len r - 1 by H2,XXREAL_0:2;
               then i >= len r - 1 + 1 by INT_1:7;
               hence contradiction by H1,ALGSEQ_1:8;
               end;
           hence r.i = q.i by A2;
           hence i in Support q by H1,POLYNOM1:def 4;
           end;
        LC r is non zero; then
        r.(len r - 1) <> 0.F by K,RATFUNC1:def 6; then
        r.(deg r) <> 0.F by HURWITZ:def 2; then
        deg r in Support r by POLYNOM1:def 4; then
        deg r in Support q by H; then
        consider k1 being Nat such that
        A6: deg r = p * k1 by A1,NAT_D:def 3;
        len r - 1 < len q - 1 by A2,XREAL_1:9; then
        deg r < len q - 1 by HURWITZ:def 2; then
        deg r < deg q by HURWITZ:def 2; then
        (p * k1) / p < (p * k) / p by A1,A6,XREAL_1:74; then
        k1 * (p / p) < (p * k) / p; then
        k1 * 1 < k * (p / p) by XCMPLX_1:60; then
        A7: k1 < k * 1 by XCMPLX_1:60;
        now let i be Nat;
            assume K: i in Support r; then
            L: i in Support q by H;
            hence p divides i by A1;
            consider a being Element of F such that
            M: a|^p = q.i by L,A1;
            r.i = a|^p by M,K,H;
            hence ex a being Element of F st a|^p = r.i;
            end; then
        consider r2 being Element of the carrier of Polynom-Ring F such that
        A8: r2|^p = r by A6,A7,IV;
        (r1 + r2)|^p = r1|^p + r2|^p by fresh
                    .= q by A2,A5,A8,POLYNOM3:def 10;
        hence
           ex r being Element of the carrier of Polynom-Ring F st r|^p = q;
        end;
      end;
    hence P[k];
    end;
I: for k being Nat holds P[k] from NAT_1:sch 4(IS);
LC q is non zero; then
q.(len q -' 1) <> 0.F by RATFUNC1:def 6; then
len q -' 1 in Support q by POLYNOM1:def 4; then
consider k being Nat such that H: len q -' 1 = p * k by AS,NAT_D:def 3;
q <> 0_.(F); then
len q <> 0 by POLYNOM4:5; then
len q -' 1 = len q - 1 by XREAL_0:def 2; then
deg q = len q -' 1 by HURWITZ:def 2;
hence thesis by AS,H,I;
end;
end;
