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;
