
theorem Th41:
  for n being non zero Element of NAT for x being Element of
F_Complex holds eval(unital_poly(F_Complex,n), x) = (power F_Complex).(x,n) - 1
proof
  1 - 1 = 0;
  then
A1: 1 -'1 = 0 by XREAL_1:233;
  reconsider z2=1_F_Complex as Element of COMPLEX by COMPLFLD:def 1;
  let n be non zero Element of NAT, x be Element of F_Complex;
  set p = unital_poly(F_Complex,n);
  consider F being FinSequence of F_Complex such that
A2: eval(p,x) = Sum F and
A3: len F = len p and
A4: for i being Element of NAT st i in dom F holds F.i = p.(i-'1) * (
  power F_Complex).(x,i-'1) by POLYNOM4:def 2;
A5: 0+1 < n+1 by XREAL_1:8;
  then
A6: 1 < len F by A3,Th40;
A7: len F=n+1 by A3,Th40;
  then len F -1=n;
  then
A8: len F -'1 = n by A5,XREAL_1:233;
  len F - 1 + 1 = len F;
  then
A9: len F-'1+1 = len F by A6,XREAL_1:233;
A10: p.0 = -1_F_Complex by Th38;
  set xn = (power F_Complex).(x,n);
  set null = (len F-'1) |-> (0.F_Complex);
A11: len null = len F -'1 by CARD_1:def 7;
  set tR2 = null^<*xn*>;
  set tR1 = <*-1_F_Complex*>^null;
A12: dom F = Seg len F by FINSEQ_1:def 3;
  reconsider R1=tR1 as Tuple of len F, the carrier of F_Complex
  by A9;
  reconsider R1 as Element of (len F)-tuples_on the carrier of F_Complex
           by FINSEQ_2:131;
  reconsider R2=tR2 as Tuple of len F, the carrier of F_Complex
  by A9;
  reconsider R2 as Element of (len F)-tuples_on the carrier of F_Complex
           by FINSEQ_2:131;
A13: for i being Element of NAT st i in dom null holds null.i = 0.F_Complex
  proof
    let i be Element of NAT;
    assume i in dom null;
    then i in Seg (len F-'1) by FUNCOP_1:13;
    hence thesis by FUNCOP_1:7;
  end;
A14: for i being Nat st i <> 1 & i in dom R1 holds R1.i = 0.F_Complex
  proof
    let i be Nat such that
A15: i <> 1 and
A16: i in dom R1;
A17: dom <*-1_F_Complex*> = Seg 1 by FINSEQ_1:def 8;
    now
      assume i in dom <*-1_F_Complex*>;
      then 1<=i & i<=1 by A17,FINSEQ_1:1;
      hence contradiction by A15,XXREAL_0:1;
    end;
    then consider j being Nat such that
A18: j in dom null and
A19: i = len <*-1_F_Complex*> + j by A16,FINSEQ_1:25;
    null.j = 0.F_Complex by A13,A18;
    hence thesis by A18,A19,FINSEQ_1:def 7;
  end;
  len tR2 = len null + len <*xn*> by FINSEQ_1:22;
  then
A20: len tR2 = len F by A11,A9,FINSEQ_1:39;
A21: for i being Element of NAT st i in dom R2 & i <> len F holds R2.i = 0.
  F_Complex
  proof
    let i be Element of NAT such that
A22: i in dom R2 and
A23: i <> len F;
A24: dom R2 = Seg len F by A20,FINSEQ_1:def 3;
    then i <= len F by A22,FINSEQ_1:1;
    then i < len F-'1+1 by A9,A23,XXREAL_0:1;
    then
A25: i <= len F-'1 by NAT_1:13;
    1 <= i by A22,A24,FINSEQ_1:1;
    then i in Seg (len F-'1) by A25,FINSEQ_1:1;
    then
A26: i in dom null by A11,FINSEQ_1:def 3;
    then R2.i = null.i by FINSEQ_1:def 7;
    hence thesis by A13,A26;
  end;
  len R1 = len F by CARD_1:def 7;
  then
A27: dom R1 = Seg len F by FINSEQ_1:def 3;
  len R2 = len F by CARD_1:def 7;
  then
A28: dom R2 = Seg len F by FINSEQ_1:def 3;
A29: R1.1 = -1_F_Complex by FINSEQ_1:41;
A30: len (R1+R2) = len F by CARD_1:def 7;
  then
A31: dom (R1+R2) = Seg len F by FINSEQ_1:def 3;
A32: R2.(len F) = (power F_Complex).(x,n) by A11,A9,FINSEQ_1:42;
  for k being Nat st k in dom (R1+R2) holds (R1+R2).k = F.k
  proof
    let k be Nat such that
A33: k in dom (R1+R2);
A34: k in Seg len F by A30,A33,FINSEQ_1:def 3;
A35: k in dom F by A31,A33,FINSEQ_1:def 3;
A36: 1 <= k by A31,A33,FINSEQ_1:1;
A37: (-1_F_Complex)*(1_F_Complex) = -1_F_Complex;
    now
      per cases;
      suppose
A38:    k = 1;
        then R2.k = 0.F_Complex by A6,A21,A28,A34;
        then
A39:    (R1+R2).1 = (-1_F_Complex)+0.F_Complex by A29,A33,A38,FVSUM_1:17;
        F.1 = p.0 * (power F_Complex).(x,0) by A4,A1,A35,A38
          .= -1_F_Complex by A10,A37,GROUP_1:def 7;
        hence thesis by A38,A39,COMPLFLD:7;
      end;
      suppose
A40:    k <> 1;
        now
          per cases;
          suppose
A41:        k = len F;
            len F <> 0 by A3,A5,Th40;
            then
A42:        F.(len F) = p.(len F-'1)*(power F_Complex).(x,len F-'1) by A4,A12,
FINSEQ_1:3
              .= 1_F_Complex*(power F_Complex).(x,n) by A8,Th38
              .= (power F_Complex).(x,n);
            R1.(len F) = 0.F_Complex by A6,A14,A27,A34,A41;
            then (R1+R2).(len F) = 0.F_Complex + (power F_Complex).(x, n) by
A32,A33,A41,FVSUM_1:17
              .= (power F_Complex).(x,n) by COMPLFLD:7;
            hence thesis by A41,A42;
          end;
          suppose
A43:        k <> len F;
A44:        now
              assume k-'1 = n;
              then k - 1 = n by A36,XREAL_1:233;
              hence contradiction by A7,A43;
            end;
            1 < k by A36,A40,XXREAL_0:1;
            then 1+-1 < k+-1 by XREAL_1:8;
            then 1-1 < k-1;
            then 0 < k-'1 by A36,XREAL_1:233;
            then p.(k-'1) = 0.F_Complex by A44,Th39;
            then
A45:        F.k = 0.F_Complex * (power F_Complex).(x,k-'1) by A4,A35;
A46:        R2.k = 0.F_Complex by A21,A28,A34,A43;
            R1.k = 0.F_Complex by A14,A27,A34,A40;
            then (R1+R2).k = 0.F_Complex + 0.F_Complex by A33,A46,FVSUM_1:17;
            hence thesis by A45,COMPLFLD:7;
          end;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  then
A47: (R1+R2) = F by A12,A31,FINSEQ_1:13;
  Sum null = 0.F_Complex by MATRIX_3:11;
  then Sum R1 = -1_F_Complex + 0.F_Complex & Sum R2 = 0.F_Complex + xn by
FVSUM_1:71,72;
  then -z2 = -1_F_Complex & Sum F = -1_F_Complex+(power F_Complex).(x,n) by A47
,COMPLFLD:2,7,FVSUM_1:76;
  hence thesis by A2,COMPLFLD:8;
end;
