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 Th51:
  th > 0 implies exp_R.th >=1
proof
  assume
A1: th >0;
A2: for n holds Partial_Sums(th rExpSeq).n>=1
  proof
    defpred X[Nat] means Partial_Sums(th rExpSeq).$1>=1;
 Partial_Sums(th rExpSeq).0=(th rExpSeq).0 by SERIES_1:def 1
      .=th |^ 0 /(0!) by Def5
      .=1 by NEWTON:4,12;
then A3: X[0];
A4: for n st X[n] holds X[n+1]
    proof
      let n;
      assume
A5:   Partial_Sums(th rExpSeq).n>=1;
A6:   Partial_Sums(th rExpSeq).(n+1)
      =Partial_Sums(th rExpSeq).n+(th rExpSeq).(n+1) by SERIES_1:def 1
        .=Partial_Sums(th rExpSeq).n+th |^(n+1)/((n+1)!) by Def5;
   th |^(n+1) >0 & (n+1)!>0 by A1,PREPOWER:6;
then    Partial_Sums(th rExpSeq).n+th |^(n+1)/((n+1)!)>Partial_Sums(th
      rExpSeq).n by XREAL_1:29;
      hence thesis by A5,A6,XXREAL_0:2;
    end;
 for n holds X[n] from NAT_1:sch 2(A3,A4);
    hence thesis;
  end;
th rExpSeq is summable by Th44;
then A7: Partial_Sums(th rExpSeq) is convergent by SERIES_1:def 2;
 lim(Partial_Sums(th rExpSeq))=Sum(th rExpSeq) by SERIES_1:def 3;
then  Sum(th rExpSeq)>=1 by A2,A7,PREPOWER:1;
  hence thesis by Def22;
end;
