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 Th52:
  th < 0 implies 0<exp_R.th & exp_R.th <=1
proof
  assume th <0;
then A1: exp_R.(-th)>=1 by Th51;
A2: exp_R.(-th)*exp_R.th=exp_R.(-th+th) by Lm10
    .=1 by Lm11;
then A3: exp_R.th=1/(exp_R.(-th)) by XCMPLX_1:73;
  thus 0<exp_R.th by A1,A2;
  thus thesis by A1,A3,XREAL_1:211;
end;
