
theorem Th28:
for p being real Polynomial of F_Complex
for x being Element of F_Complex st Re(x) = 0
holds Im(eval(even_part(p),x)) = 0
proof
let p be real Polynomial of F_Complex;
let x be Element of F_Complex;
defpred P[Nat] means
   for p being Polynomial of F_Complex st p is real & len p = $1 holds
   (for x being Element of F_Complex st Re(x) = 0
    holds Im(eval(even_part(p),x)) = 0);
A1: now let k be Nat;
   assume A2: for n being Nat st n < k holds P[n];
   now let p be Polynomial of F_Complex;
     assume A3: p is real & len p = k;
     now per cases by NAT_1:14;
     case k = 0;
       then p = 0_.(F_Complex) by A3,POLYNOM4:5;
       then A4: even_part(p) = 0_.(F_Complex) by Th7;
       thus for x being Element of F_Complex st Re(x) = 0
            holds Im(eval(even_part(p),x)) = 0
            proof
            let x be Element of F_Complex;
            assume Re(x) = 0;
            eval(even_part(p),x) = 0.F_Complex by A4,POLYNOM4:17
                                .= 0 by COMPLFLD:def 1;
            hence thesis by COMPLEX1:4;
            end;
       end;
     case A5: k >= 1;
       set LMp = Leading-Monomial(p);
       LMp + (p-LMp) = (LMp + - LMp) + p by POLYNOM3:26
                    .= (LMp - LMp) + p
                    .= 0_.(F_Complex) + p by POLYNOM3:29
                    .= p by POLYNOM3:28;
       then A6: even_part(p) = even_part(LMp) + even_part(p-LMp) by Th15;
       thus for x being Element of F_Complex st Re(x) = 0
            holds Im(eval(even_part(p),x)) = 0
            proof
            let x be Element of F_Complex;
            assume A7: Re(x) = 0;
            consider t being Polynomial of F_Complex such that
            A8: len t < len p & p = t + Leading-Monomial(p) &
               for n being Element of NAT st n < len p-1 holds t.n = p.n
               by A5,A3,POLYNOM4:16;
            A9: p - LMp = t + (LMp - LMp) by A8,POLYNOM3:26
                       .= t + 0_.(F_Complex) by POLYNOM3:29
                       .= t by POLYNOM3:28;
             now let n be Nat;
A10:              n in NAT by ORDINAL1:def 12;
                now per cases;
                case n < len p-1;
                  then t.n = p.n by A8,A10;
                  hence t.n is Real by A3;
                  end;
                case A11: n >= len p-1;
                  reconsider lp = len p-1 as Element of NAT by A5,A3,INT_1:5;
                  len t < lp + 1 by A8;
                  then lp >= len t by NAT_1:13;
                  then t.n = 0.F_Complex by ALGSEQ_1:8,A11,XXREAL_0:2;
                  hence t.n is Real by COMPLFLD:def 1;
                  end;
                end;
                hence t.n is Real;
                end;
            then A12: t is real;
            A13: Im(eval(even_part(LMp),x)) = 0
               proof
               now per cases;
               case deg p is odd;
                  then even_part(LMp) = 0_.(F_Complex) by Th18;
                  then eval(even_part(LMp),x) = 0.F_Complex by POLYNOM4:17
                                             .= 0 by COMPLFLD:def 1;
                  hence thesis by COMPLEX1:4;
                  end;
               case A14: deg p is even;
                  then A15: eval(even_part(LMp),x)
                        = eval(LMp,x) by Th17
                       .= p.(len p-'1) * (power F_Complex).(x,len p-'1)
                          by POLYNOM4:22;
                  set z1 = p.(len p-'1), z2 = (power F_Complex).(x,len p-'1);
                  len p -' 1 = deg p by A3,A5,XREAL_1:233;
                  then A16: Im z2 = 0 by A7,A14,Th5;
                  z1 in REAL by A3,XREAL_0:def 1;
                  then A17: Im z1 = 0 by COMPLEX1:def 2;
                  thus Im(eval(even_part(LMp),x))
                     = Re z1 * Im z2 + Re z2 * Im z1 by A15,COMPLEX1:9
                    .= 0 by A16,A17;
                  end;
               end;
               hence thesis;
               end;
            thus Im(eval(even_part(p),x))
                = Im(eval(even_part(LMp),x) + eval(even_part(p-LMp),x))
                  by A6,POLYNOM4:19
               .= 0 + Im(eval(even_part(p-LMp),x)) by A13,COMPLEX1:8
               .= 0 by A12,A8,A9,A7,A3,A2;
            end;
       end;
     end;
     hence for x being Element of F_Complex st Re(x) = 0
           holds Im(eval(even_part(p),x)) = 0;
     end;
   hence P[k];
   end;
A18: for k being Nat holds P[k] from NAT_1:sch 4(A1);
consider n being Nat such that A19: len p = n;
thus thesis by A18,A19;
end;
