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 Th39:
  0<=th & th <=1 & n<=k implies th |^ k <= th |^ n
proof
  assume that
A1: 0<=th and
A2: th <=1 and
A3: n<=k;
 for m holds th GeoSeq.(m+1)<=th GeoSeq.m& th GeoSeq.m >=0
  proof
    let m;
    defpred X[Nat] means
    th GeoSeq.($1+1)<=th GeoSeq.$1 &th GeoSeq.$1 >=0;
 th GeoSeq.(0+1)=th GeoSeq.0 * th by PREPOWER:3
      .= 1*th by PREPOWER:3
      .=th;
then A4: X[0] by A2,PREPOWER:3;
A5: for m st X[m] holds X[m+1]
    proof
      let m;
      assume that
      th GeoSeq.(m+1)<=th GeoSeq.m and
A6:   th GeoSeq.m >=0;
      th GeoSeq.(m+1+1)=th GeoSeq.(m+1)*th &
      th GeoSeq.(m+1)=th GeoSeq.(m)*th by PREPOWER:3;
      hence thesis by A1,A2,A6,XREAL_1:153;
    end;
    for m holds X[m] from NAT_1:sch 2(A4,A5);
    hence thesis;
  end;
then A7: th GeoSeq is non-increasing;
   th |^ k=th GeoSeq. k & th |^ n=th GeoSeq. n by PREPOWER:def 1;
  hence thesis by A3,A7,SEQM_3:8;
end;
