reserve q,th,r for Real,
  a,b,p for Real,
  w,z for Complex,
  k,l,m,n,n1,n2 for Nat,
  seq,seq1,seq2,cq1 for Complex_Sequence,
  rseq,rseq1,rseq2 for Real_Sequence,
  rr for set,
  hy1 for 0-convergent non-zero Real_Sequence;
reserve d for Real;
reserve th,th1,th2 for Real;

theorem Th33:
  for th holds (th*<i>) |^ (2*k)=(-1)|^ k* (th |^ (2*k))
  & (th*<i>) |^ (2*k+1)=(-1)|^ k* (th |^ (2*k+1))*<i>
proof
  let th;
A1: (-1)|^ 0* (th |^ (2*0))*<i>=(-1)|^ 0* th GeoSeq.0*<i> by PREPOWER:def 1
    .=(-1)|^ 0*(1)*<i> by PREPOWER:3
    .=(-1)GeoSeq.0*<i> by PREPOWER:def 1
    .=1*<i> by PREPOWER:3;
  defpred X[Nat] means
  (th*<i>) |^ (2*$1)=(-1)|^ $1* (th |^ (2*$1))
  & (th*<i>) |^ (2*$1+1)=(-1)|^ $1* (th |^ (2*$1+1))*<i>;
 (-1)|^ 0 * (th |^ (2*0+1))*<i>=(-1)|^ 0 * (th GeoSeq.(2*0+1))*<i>
  by PREPOWER:def 1
    .= (-1)|^ 0* (th GeoSeq.0*th)*<i> by PREPOWER:3
    .=(-1)|^ 0* (1*th)*<i> by PREPOWER:3
    .= ((-1)GeoSeq.0)* th *<i> by PREPOWER:def 1
    .=(1)* th *<i> by PREPOWER:3
    .=(th*<i>);
then A2: X[0] by A1,COMSEQ_3:def 1;
A3: for k st X[k] holds X[k+1]
  proof
    let k;
    assume that
A4: (th*<i>) |^ (2*k)=(-1)|^ k* (th |^ (2*k)) and
A5: (th*<i>) |^ (2*k+1)=(-1)|^ k* (th |^ (2*k+1))*<i>;
A6: (th*<i>) |^ (2*(k+1))= ((th*<i>) |^ 2)|^(k+1) by Th32
      .=((th*<i>) |^ 2)|^ k * ((th*<i>) |^ 2) by COMSEQ_3:def 1
      .= (-1)|^ k* (th |^ (2*k))* ((th*<i>)GeoSeq.(1+1)) by A4,Th32
      .=(-1)|^ k* (th |^ (2*k))* ((th*<i>)GeoSeq.(0+1)*(th*<i>))
    by COMSEQ_3:def 1
      .=(-1)|^ k* (th |^ (2*k))* ((th*<i>)GeoSeq.0*(th*<i>)
    *(th*<i>)) by COMSEQ_3:def 1
      .= (-1)|^ k* (th |^ (2*k))* (1r*(th*<i>)*(th*<i>))
    by COMSEQ_3:def 1
      .=(-1)|^ k*(-1)* (th |^ (2*k))*th*th
      .=(-1)GeoSeq.k*(-1)* (th |^ (2*k))*th*th by PREPOWER:def 1
      .=(-1)GeoSeq.(k+1)* (th |^ (2*k))*th*th by PREPOWER:3
      .=(-1)|^(k+1)* (th |^ (2*k))*th*th by PREPOWER:def 1
      .=(-1)|^(k+1)* (th GeoSeq.(2*k))*th*th by PREPOWER:def 1
      .= (-1)|^(k+1)* (th GeoSeq.(2*k)*th)*th
      .=(-1)|^(k+1)* (th GeoSeq.(2*k+1))*th by PREPOWER:3
      .=(-1)|^(k+1)* (th GeoSeq.(2*k+1)*th)
      .=(-1)|^(k+1)* (th GeoSeq.(2*k+1+1)) by PREPOWER:3
      .=(-1)|^(k+1)* (th |^ (2*(k+1))) by PREPOWER:def 1;
 (th*<i>) |^ (2*(k+1)+1)=
    (th*<i>)GeoSeq.(2*k+1+1)*(th*<i>) by COMSEQ_3:def 1
      .=(th*<i>)GeoSeq.(2*k+1)*(th*<i>)*(th*<i>) by COMSEQ_3:def 1
      .=(-1)|^ k*(-1)* (th |^ (2*k+1))*th*th*<i> by A5
      .=(-1)GeoSeq.k*(-1)* (th |^ (2*k+1))*th*th*<i> by PREPOWER:def 1
      .=(-1)GeoSeq.(k+1)* (th |^ (2*k+1))*th*th*<i> by PREPOWER:3
      .=(-1)|^ (k+1)* (th |^ (2*k+1))*th*th*<i> by PREPOWER:def 1
      .=(-1)|^ (k+1)* (th GeoSeq.(2*k+1))*th*th*<i> by PREPOWER:def 1
      .=(-1)|^ (k+1)* (th GeoSeq.(2*k+1)*th)*th*<i>
      .=(-1)|^ (k+1)* (th GeoSeq.(((2*k+1)+1)))*th*<i> by PREPOWER:3
      .=(-1)|^ (k+1)* (th GeoSeq.((2*k+1+1))*th)*<i>
      .= (-1)|^ (k+1)* (th GeoSeq.(2*(k+1)+1))*<i> by PREPOWER:3
      .=(-1)|^ (k+1)*(th |^ (2*(k+1)+1))*<i> by PREPOWER:def 1;
    hence thesis by A6;
  end;
 for k holds X[k] from NAT_1:sch 2(A2,A3);
  hence thesis;
end;
