
theorem Th47:
  for x,z being Element of F_Complex holds eval(rpoly(1,z)*',x) = -x - (z*')
proof
  let x,z be Element of F_Complex;
  set p = rpoly(1,z)*';
  consider F be FinSequence of F_Complex such that
A1: eval(p,x) = Sum F and
A2: len F = len p and
A3: for n be Element of NAT st n in dom F holds F.n = p.(n-'1) * (power
  F_Complex).(x,n-'1) by POLYNOM4:def 2;
A4: deg p = deg rpoly(1,z) by Th42
    .= 1 by Th27;
  then
A5: F = <*F.1,F.2*> by A2,FINSEQ_1:44
    .= <*F.1*>^<*F.2*>;
  len p = 1 + 1 by A4;
  then 1 in Seg(len F) by A2;
  then
A6: 1 in dom F by FINSEQ_1:def 3;
A7: 2-'1 = 2-1 by XREAL_0:def 2;
  2 in Seg(len F) by A2,A4;
  then 2 in dom F by FINSEQ_1:def 3;
  then
A8: F.2 = p.1 * (power F_Complex).(x,1+0) by A3,A7
    .= p.1 * ((power F_Complex).(x,0) * x) by GROUP_1:def 7
    .= p.1 * (1_F_Complex * x) by GROUP_1:def 7
    .= p.1 * x
    .= (power(F_Complex).(-1_F_Complex,1) * (rpoly(1,z).1)*') * x by Def9
    .= (power(F_Complex).(-1_F_Complex,1) * (1_F_Complex)) * x by Lm10,
COMPLFLD:49
    .= power(F_Complex).(-1_F_Complex,1+0) * x
    .= (power(F_Complex).(-1_F_Complex,0) * (-1_F_Complex)) * x by
GROUP_1:def 7
    .= (1_F_Complex * (-1_F_Complex)) * x by GROUP_1:def 7
    .= (-(1_F_Complex)) * x
    .= -(1_F_Complex * x) by VECTSP_1:9
    .= -x;
A9: rpoly(1,z).0 = -power(F_Complex).(z,1+0) by Lm10
    .= -(power(F_Complex).(z,0) * z) by GROUP_1:def 7
    .= -(1_F_Complex * z) by GROUP_1:def 7
    .= -z;
  1-'1 = 1-1 by XREAL_0:def 2;
  then F.1 = p.0 * (power F_Complex).(x,0) by A3,A6
    .= p.0 * 1_F_Complex by GROUP_1:def 7
    .= p.0
    .= power(F_Complex).(-1_F_Complex,0) * (-z)*' by A9,Def9
    .= 1_F_Complex * (-z)*' by GROUP_1:def 7
    .= (-z)*'
    .= -(z*') by COMPLFLD:52;
  hence eval(p,x) = Sum(<*-(z*')*>) + Sum(<*-x*>) by A1,A5,A8,RLVECT_1:41
    .= Sum(<*-(z*')*>) + -x by RLVECT_1:44
    .= -(z*') + -x by RLVECT_1:44
    .= -x - (z*') by RLVECT_1:def 11;
end;
