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 Th50:
  for p being INT -valued Polynomial of F_Real
  for e being Element of F_Real st e is_a_root_of p
  for k,l being Integer st l <> 0 & e = k/l & k,l are_coprime
  holds k divides p.0 & l divides LC p
  proof
    let p be INT -valued Polynomial of F_Real;
    let e be Element of F_Real such that
A1:  e is_a_root_of p;
    let k,l be Integer such that
A2: l <> 0 & e = k/l & k,l are_coprime;
    consider F being FinSequence of F_Real such that
A3:  0.F_Real = Sum F & len F = len p & for n be Element of NAT st n in
    dom F holds F.n = p.(n-'1) * (power F_Real).(e,n-'1) by A1,POLYNOM4:def 2;
    per cases;
    suppose len F = 0; then
A4:   p = <%0.F_Real%> by A3,ALGSEQ_1:14;
      then p.0=0.F_Real by ALGSEQ_1:16;
      hence k divides p.0 by INT_2:12;
      p=0_.F_Real by A4,POLYNOM5:34;
      then LC p = 0.F_Real by FUNCOP_1:7;
      hence l divides LC p by INT_2:12;
    end;
    suppose
A5:   len F > 0;
      set n = len p;
A6:   n >= 1 by A5,A3,NAT_1:14;
      reconsider n1=n-1 as Element of NAT by NAT_1:20,A5,A3;
A7:   n-'1 = n1 by A5,A3,NAT_1:14,XREAL_1:233;
A8:  l|^(n-'1) <> 0 by A2,CARD_4:3;
      reconsider k1=k,l1 = l as Element of F_Real by XREAL_0:def 1;
      set ln=(power F_Real).(l1,n1);
      set G=ln*F;
      reconsider FF=F as Element of (len F)-tuples_on the carrier of F_Real
      by FINSEQ_2:92;
      set GG=ln*FF;
A9:   len GG = len F by FINSEQ_2:132;
      then
A10:   dom G = dom F by FINSEQ_3:29;
A11:    Sum G = ln * Sum F by FVSUM_1:73;
      rng G c= INT
      proof
        let o be object;
        assume o in rng G;
        then consider b being object such that
A12:      b in dom G & o=G.b by FUNCT_1:def 3;
        reconsider b as Element of NAT by A12;
        b in Seg n by A12,A9,A3,FINSEQ_1:def 3;
        then 1 <= b <= n & b-'1 <= b by FINSEQ_1:1,NAT_D:35;
        then b-'1 = b-1 & b-1 <= n1 by XREAL_1:233,XREAL_1:9;
        then consider c being Nat such that
A13:      n1=b-'1+c by NAT_1:10;
        rng F c= the carrier of F_Real; then
        reconsider a9=F.b as Element of F_Real by A12,A10,FUNCT_1:3;
A14:    l|^(b-'1) <> 0 by A2,CARD_4:3;
        b in dom (ln*F) & a9 = F.b implies (ln*F).b = ln*a9 by FVSUM_1:50;
        then
        G.b = ln*(p.(b-'1) * (power F_Real).(e,b-'1)) by A3,A12,A10
        .= p.(b-'1) * ((power F_Real).(l1,n1) * (power F_Real).(e,b-'1))
        .= p.(b-'1) * ((l1|^n1) * (power F_Real).(e,b-'1)) by Th48
        .= p.(b-'1) * ((l1|^n1) * ((k/l)|^(b-'1))) by A2,Th48
        .= p.(b-'1) * ((l|^n1) * ((k|^(b-'1))/(l|^(b-'1)))) by PREPOWER:8
        .= p.(b-'1) * (k|^(b-'1)) * ((l|^n1)/(l|^(b-'1)))
        .= p.(b-'1) * (k|^(b-'1)) * ((l|^c)*(l|^(b-'1))/(l|^(b-'1)))
        by A13,NEWTON:8
        .= p.(b-'1) * (k|^(b-'1)) * ((l|^c)*((l|^(b-'1))/(l|^(b-'1))))
        .= p.(b-'1) * (k|^(b-'1)) * ((l|^c) * 1) by XCMPLX_1:60,A14
        .= p.(b-'1) * (k|^(b-'1)) * (l|^c);
        hence o in INT by INT_1:def 2,A12;
      end;
      then reconsider G1=G as non empty INT -valued FinSequence
      by A9,A5,RELAT_1:def 19;
A15:   1 in dom G by A9,A6,A3,FINSEQ_3:25;
A16:  Sum G1 = Sum G by Th49;
A17: Sum G1 = 0 by A3,A11,Th49;
      reconsider Gn0=G1/^1 as INT -valued FinSequence;
      G = <*G/.1*>^Gn0 by FINSEQ_5:29;
      then Sum Gn0 + G/.1 = 0 by RVSUM_1:76,A16,A11,A3;
      then Sum Gn0 + G.1 = 0 by A15,PARTFUN1:def 6;
      then
A18:   Sum Gn0 = - G1.1;
      rng F c= the carrier of F_Real; then
      reconsider a9=F.1 as Element of F_Real by A15,A10,FUNCT_1:3;
A19:   G1.1 = ln * a9 by FVSUM_1:50,A15
      .= ln * (p.(1-'1) * (power F_Real).(e,1-'1)) by A6,A3,FINSEQ_3:25
      .= p.(1-'1) * ln * (power F_Real).(e,1-'1)
      .= p.(1-'1) * ln * (power F_Real).(e,0) by XREAL_1:232
      .= p.0 * ln * (power F_Real).(e,0) by XREAL_1:232
      .= p.0 * ln * 1_F_Real by GROUP_1:def 7
      .= p.0 * (l|^n1) by Th48;
      for i being Nat st i in dom Gn0 holds k divides Gn0.i
      proof
        let i be Nat;
        assume
A20:    i in dom Gn0; then
A21:    1+i in dom G1 by FINSEQ_5:26;
        rng F c= the carrier of F_Real; then
        reconsider a9=F.(1+i) as Element of F_Real by A21,A10,FUNCT_1:3;
A22:    l|^i <> 0 by A2,CARD_4:3;
A23:    1 <= i <= len Gn0 by A20,FINSEQ_3:25;
        i+1 in Seg n by A3,A21,A9,FINSEQ_1:def 3;
        then i+1 <= n by FINSEQ_1:1;
        then i+1-1 <= n-1 by XREAL_1:9;
        then consider d being Nat such that
A24:    n1=i+d by NAT_1:10;
        Gn0.i = (G/^1)/.i by A20,PARTFUN1:def 6
        .= G/.(1+i) by FINSEQ_5:27,A20
        .= G.(1+i) by A21,PARTFUN1:def 6
        .= ln * a9 by FVSUM_1:50,A21
        .= ln*(p.(1+i-'1) * (power F_Real).(e,1+i-'1)) by A3,A21,A10
        .= ln*(p.i * (power F_Real).(e,1+i-'1)) by NAT_D:34
        .= ln*(p.i * (power F_Real).(e,i)) by NAT_D:34
        .= p.i * ((power F_Real).(l1,n1) * (power F_Real).(e,i))
        .= p.i * ((l1|^n1) * (power F_Real).(e,i)) by Th48
        .= p.i * ((l1|^n1) * (e|^i)) by Th48
        .= p.i * ((l|^n1) * ((k|^i)/(l|^i))) by A2,PREPOWER:8
        .= p.i * (k|^i) * ((l|^(d+i)/(l|^i))) by A24
        .= p.i * (k|^i) * ((l|^d)*(l|^i)/(l|^i)) by NEWTON:8
        .= p.i * (k|^i) * ((l|^d)*((l|^i)/(l|^i)))
        .= p.i * (k|^i) * ((l|^d) * 1) by XCMPLX_1:60,A22
        .= p.i * (l|^d) * (k|^i);
        hence k divides Gn0.i by A23,Th4,INT_2:2;
      end;
      then k divides G1.1 by A18,NEWTON04:80,INT_2:10;
      hence k divides p.0 by A19,A2,WSIERP_1:10,INT_2:25;
      reconsider Gn1=G1|(Seg n1) as INT -valued FinSequence by FINSEQ_1:15;
A25:   len GG = len F by FINSEQ_2:132; then
A26:   len G1 = n1+1 by A3;
      G1 = Gn1^<*G1.(n1+1)*> by A25,A3,FINSEQ_3:55;
      then Sum Gn1 + G1.(n1+1) = 0 by RVSUM_1:74,A17; then
A27:   Sum Gn1 = - G1.(n1+1) .= - G1.n;
A28:   n in dom F by FINSEQ_3:25,A6,A3;
      rng F c= the carrier of F_Real; then
      reconsider a9=F.n as Element of F_Real by A28,FUNCT_1:3;
      n in dom G1 by A25,A3,FINSEQ_3:25,A6; then
A29:   G1.n = ln * a9 by FVSUM_1:50
      .= ln * (p.(n-'1) * (power F_Real).(e,n-'1)) by A6,A3,FINSEQ_3:25
      .= p.(n-'1) * ((power F_Real).(l1,n1) * (power F_Real).(e,n-'1))
      .= p.(n-'1) * ((l1|^n1) * (power F_Real).(e,n-'1)) by Th48
      .= p.(n-'1) * ((l1|^n1) * (e|^(n-'1))) by Th48
      .= p.(n-'1) * ((l|^n1) * ((k|^(n-'1))/(l|^(n-'1)))) by A2,PREPOWER:8
      .= p.(n-'1) * (k|^(n-'1)) * ((l|^(n-'1))/(l|^(n-'1))) by A7
      .= p.(n-'1) * (k|^(n-'1)) * 1 by A8,XCMPLX_1:60
      .= LC p * (k|^(n-'1));
      for i being Nat st i in dom Gn1 holds l divides Gn1.i
      proof
        let i be Nat;
        assume
A30:    i in dom Gn1;
        then i in Seg n1 by A26,FINSEQ_3:54; then
A31:     1 <= i <= n1 & i-'1 <= i by FINSEQ_1:1,NAT_D:35;
        then consider d being Nat such that
A32:      n1=i-'1+d by XXREAL_0:2,NAT_1:10;
        i-i <= n-1-i by A31,XREAL_1:9; then
A33:     0+1<=n-i-1+1 by XREAL_1:6;
A34:     n-1=i-1+d by A32,A31,XREAL_1:233;
A35:     Gn1.i = G1.i by A30,FUNCT_1:47;
A36:     dom Gn1 c= dom G1 by RELAT_1:60;
        rng F c= the carrier of F_Real; then
        reconsider a9=F.i as Element of F_Real by A36,A10,A30,FUNCT_1:3;
A37:    l|^(i-'1) <> 0 by A2,CARD_4:3;
        G1.i = ln * a9 by A36,A30,FVSUM_1:50
        .= ln*(p.(i-'1) * (power F_Real).(e,i-'1)) by A3,A36,A30,A10
        .= p.(i-'1) * ((power F_Real).(l1,n1) * (power F_Real).(e,i-'1))
        .= p.(i-'1) * ((l1|^n1) * (power F_Real).(e,i-'1)) by Th48
        .= p.(i-'1) * ((l1|^n1) * (e|^(i-'1))) by Th48
        .= p.(i-'1) * ((l|^n1) * ((k|^(i-'1))/(l|^(i-'1)))) by A2,PREPOWER:8
        .= p.(i-'1) * (k|^(i-'1)) * ((l|^n1)/(l|^(i-'1)))
        .= p.(i-'1) * (k|^(i-'1)) * ((l|^d)*(l|^(i-'1))/(l|^(i-'1)))
        by A32,NEWTON:8
        .= p.(i-'1) * (k|^(i-'1)) * ((l|^d)*((l|^(i-'1))/(l|^(i-'1))))
        .= p.(i-'1) * (k|^(i-'1)) * ((l|^d) * 1) by XCMPLX_1:60,A37
        .= p.(i-'1) * (k|^(i-'1)) * (l|^d);
        hence l divides Gn1.i by A35,A34,A33,Th4,INT_2:2;
      end;
      then l divides G1.n by A27,NEWTON04:80,INT_2:10;
      hence l divides LC p by A29,A2,WSIERP_1:10,INT_2:25;
    end;
  end;
