
theorem Th32:
  for L being add-associative right_zeroed right_complementable
left-distributive well-unital commutative non empty doubleLoopStr for z being
Element of L for k being Element of NAT st k > 1 holds rpoly(1,z) *' qpoly(k,z)
  = rpoly(k,z)
proof
  let L be add-associative right_zeroed right_complementable left-distributive
  well-unital commutative non empty doubleLoopStr;
  let z be Element of L, k be Element of NAT;
  set p = rpoly(1,z) *' qpoly(k,z), u = rpoly(k,z);
  assume
A1: k > 1;
  then reconsider k1 = k - 1 as Element of NAT by INT_1:5;
  k1 + 1 = k;
  then
A2: k1 <= k by INT_1:6;
  k1 <> k;
  then
A3: k1 < k by A2,XXREAL_0:1;
A4: now
A5: 1 - 1 >= 0;
    let i9 be object;
A6: 0 + 1 = 1;
    assume i9 in dom p;
    then reconsider i = i9 as Element of NAT;
    consider fp being FinSequence of L such that
A7: len fp = i+1 and
A8: p.i = Sum fp and
A9: for j be Element of NAT st j in dom fp holds fp.j = rpoly(1,z).(j
    -'1 ) * qpoly(k,z).(i+1-'j) by POLYNOM3:def 9;
A10: i + 1 - 2 = i - 1;
    len fp >= 1 by A7,NAT_1:11;
    then 1 in Seg(len fp);
    then
A11: 1 in dom fp by FINSEQ_1:def 3;
    then
A12: fp/.1 = fp.1 by PARTFUN1:def 6
      .= rpoly(1,z).(1-'1) * qpoly(k,z).(i+1-'1) by A9,A11
      .= rpoly(1,z).0 * qpoly(k,z).(i+1-'1) by A5,XREAL_0:def 2
      .= (-power(L).(z,1)) * qpoly(k,z).(i+1-'1) by Lm10
      .= (-power(L).(z,0)*z) * qpoly(k,z).(i+1-'1) by A6,GROUP_1:def 7
      .= (-(1_L*z)) * qpoly(k,z).(i+1-'1) by GROUP_1:def 7
      .= (-z) * qpoly(k,z).(i+1-'1)
      .= (-z) * qpoly(k,z).i by NAT_D:34;
A13: now
      let j be Element of NAT;
      assume that
A14:  j in dom fp and
A15:  j <> 1 and
A16:  j <> 2;
A17:  j in Seg(len fp) by A14,FINSEQ_1:def 3;
      now
        assume
A18:    j-'1 = 0 or j-'1 = 1;
        per cases;
        suppose
          j-1 >= 0;
          then j-'1 = j - 1 by XREAL_0:def 2;
          hence contradiction by A15,A16,A18;
        end;
        suppose
          j-1 < 0;
          then j-1+1 < 0 + 1 by XREAL_1:8;
          hence contradiction by A17,FINSEQ_1:1;
        end;
      end;
      then
A19:  rpoly(1,z).(j-'1) = 0.L by Lm11;
      fp.j = rpoly(1,z).(j-'1) * qpoly(k,z).(i+1-'j) by A9,A14;
      hence fp.j = 0.L by A19;
    end;
A20: now
A21:  1 + 1 = 2;
      consider g1,g2 being FinSequence of L such that
A22:  len g1 = 1 and
A23:  len g2 = i and
A24:  fp = g1 ^ g2 by A7,FINSEQ_2:23;
A25:  g1 = <*g1.1*> by A22,FINSEQ_1:40
        .= <*fp.1*> by A22,A24,FINSEQ_1:64
        .= <*fp/.1*> by A11,PARTFUN1:def 6;
      assume i <> 0;
      then
A26:  i+1 > 0+1 by XREAL_1:8;
      then
A27:  i >= 1 by NAT_1:13;
      then 1 in Seg len g2 by A23;
      then
A28:  1 in dom g2 by FINSEQ_1:def 3;
      1+1 <= len fp by A7,A26,NAT_1:13;
      then 2 in Seg(len fp);
      then
A29:  2 in dom fp by FINSEQ_1:def 3;
      now
        let i be Element of NAT;
        assume that
A30:    i in dom g2 and
A31:    i <> 1;
A32:    i+1<>2 by A31;
A33:    1 <= i + 1 by NAT_1:11;
A34:    i in Seg(len g2) by A30,FINSEQ_1:def 3;
        then
A35:    i <= len g2 by FINSEQ_1:1;
        len fp = 1 + len g2 by A22,A24,FINSEQ_1:22;
        then i + 1 <= len fp by A35,XREAL_1:6;
        then i + 1 in Seg(len fp) by A33;
        then
A36:    i+1 in dom fp by FINSEQ_1:def 3;
        i+1<>0+1 by A34,FINSEQ_1:1;
        then
A37:    fp.(i+1)=0.L by A13,A36,A32;
        1 <= i by A34,FINSEQ_1:1;
        then g2.i = fp.(i+1) by A22,A24,A35,FINSEQ_1:65;
        hence g2/.i = 0.L by A30,A37,PARTFUN1:def 6;
      end;
      then Sum g2 = g2/.1 by A28,POLYNOM2:3
        .= g2.1 by A28,PARTFUN1:def 6
        .= fp.2 by A27,A22,A23,A24,A21,FINSEQ_1:65
        .= fp/.2 by A29,PARTFUN1:def 6;
      hence p.i = Sum(g1) + fp/.2 by A8,A24,RLVECT_1:41
        .= fp/.1 + fp/.2 by A25,RLVECT_1:44;
    end;
A38: 2 - 1 >= 0;
A39: now
      assume i <> 0;
      then
A40:  i + 1 > 0 + 1 by XREAL_1:8;
      then i >= 1 by NAT_1:13;
      then reconsider i1 = i-1 as Element of NAT by INT_1:5;
      len fp >= 1+1 by A7,A40,NAT_1:13;
      then 2 in Seg(len fp);
      then
A41:  2 in dom fp by FINSEQ_1:def 3;
      then
A42:  fp.2 = rpoly(1,z).(2-'1) * qpoly(k,z).(i+1-'2) by A9
        .= rpoly(1,z).1 * qpoly(k,z).(i+1-'2) by A38,XREAL_0:def 2
        .= 1_L * qpoly(k,z).(i+1-'2) by Lm10
        .= qpoly(k,z).(i+1-'2)
        .= qpoly(k,z).i1 by A10,XREAL_0:def 2;
      thus fp/.2 = fp.2 by A41,PARTFUN1:def 6
        .= qpoly(k,z).(i-'1) by A42,XREAL_0:def 2;
    end;
    per cases by XXREAL_0:1;
    suppose
A43:  i < k;
      per cases;
      suppose
A44:    i = 0;
A45:    k-0-1 = k1;
A46:    k1 + 1 = k;
        fp = <*fp.1*> by A7,A44,FINSEQ_1:40
          .= <*fp/.1*> by A11,PARTFUN1:def 6;
        hence p.i9 = (-z) * qpoly(k,z).0 by A8,A12,A44,RLVECT_1:44
          .= (-z) * power(L).(z,k1) by A45,A46,Def4
          .= -(z* power(L).(z,k1)) by VECTSP_1:9
          .= - power(L).(z,k) by A46,GROUP_1:def 7
          .= u.i9 by A1,A44,Lm10;
      end;
      suppose
A47:    i > 0;
        then i + 1 > 0 + 1 by XREAL_1:6;
        then i >= 1 by NAT_1:13;
        then i - 1 >= 1 - 1 by XREAL_1:9;
        then i -' 1 = i - 1 by XREAL_0:def 2;
        then
A48:    k- (i-'1) - 1 = k - i;
        k - i > i - i by A43,XREAL_1:9;
        then reconsider ki = k - i as Element of NAT by INT_1:3;
        ki > i - i by A43,XREAL_1:9;
        then ki + 1 > 0 + 1 by XREAL_1:6;
        then ki >= 1 by NAT_1:13;
        then reconsider ki1 = k-i-1 as Element of NAT by INT_1:5;
A49:    k-1 < k-0 by XREAL_1:10;
        i-1 < k-1 by A43,XREAL_1:9;
        then
A50:    i - 1 < k by A49,XXREAL_0:2;
A51:    ki1 + 1 = ki;
        thus p.i9 = ((-z)*power(L).(z,ki1))+qpoly(k,z).(i-'1) by A12,A39,A20
,A43,A47,Def4
          .= ((-z) * power(L).(z,ki1)) + power(L).(z,ki) by A48,A50,Def4
          .= (-(z * power(L).(z,ki1))) + power(L).(z,ki) by VECTSP_1:9
          .= - power(L).(z,ki) + power(L).(z,ki) by A51,GROUP_1:def 7
          .= 0.L by RLVECT_1:5
          .= u.i9 by A43,A47,Lm11;
      end;
    end;
    suppose
A52:  i = k;
      then i - 1 >= 1 - 1 by A1,XREAL_1:9;
      then
A53:  i -' 1 = i - 1 by XREAL_0:def 2;
A54:  k - k1 - 1 = 0;
      fp/.1 = (-z) * 0.L by A12,A52,Def4
        .= 0.L;
      hence p.i9 = qpoly(k,z).(k1) by A39,A20,A52,A53,ALGSTR_1:def 2
        .= power(L).(z,0) by A3,A54,Def4
        .= 1_L by GROUP_1:def 7
        .= u.i9 by A1,A52,Lm10;
    end;
    suppose
A55:  i > k;
      then i + 1 > 0 + 1 by XREAL_1:6;
      then i >= 1 by NAT_1:13;
      then i - 1 >= 1 - 1 by XREAL_1:9;
      then
A56:  i -' 1 = i - 1 by XREAL_0:def 2;
      i >= k+1 by A55,NAT_1:13;
      then
A57:  i - 1 >= k + 1 - 1 by XREAL_1:9;
      fp/.1 = (-z) * 0.L by A12,A55,Def4
        .= 0.L;
      hence p.i9 = qpoly(k,z).(i-'1) by A39,A20,A55,ALGSTR_1:def 2
        .= 0.L by A56,A57,Def4
        .= u.i9 by A55,Lm11;
    end;
  end;
  dom p = NAT by FUNCT_2:def 1
    .= dom u by FUNCT_2:def 1;
  hence thesis by A4,FUNCT_1:2;
end;
