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 Th53:
  exp_R.th>0
proof
 now per cases;
    suppose
   th=0;
      hence thesis by Lm11;
    end;
    suppose
A1:   th<>0;
   now per cases by A1;
        suppose
       th<0;
          hence thesis by Th52;
        end;
        suppose
       th>0;
          hence thesis by Th51;
        end;
      end;
      hence thesis;
    end;
  end;
  hence thesis;
end;
