reserve Rseq, Rseq1, Rseq2 for Function of [:NAT,NAT:],REAL;
reserve rseq1,rseq2 for convergent Real_Sequence;
reserve n,m,N,M for Nat;
reserve e,r for Real;
reserve Pseq for P-convergent Function of [:NAT,NAT:],REAL;

theorem th55b:
Rseq is P-convergent &(for n,m be Nat holds Rseq.(n,m) >= r)
 implies P-lim Rseq >= r
proof
   assume a1: Rseq is P-convergent;
   assume a2: for n,m be Nat holds Rseq.(n,m) >= r;
   assume not P-lim Rseq >= r; then
   r - P-lim Rseq > 0 by XREAL_1:50; then
   consider N such that
a3: for n,m st n>=N & m>=N holds
     |. Rseq.(n,m) - P-lim Rseq.| < r - P-lim Rseq by a1,def6;
   |. Rseq.(N,N) - P-lim Rseq.| < r - P-lim Rseq by a3; then
   P-lim Rseq + (r - P-lim Rseq) > Rseq.(N,N) by RINFSUP1:1;
   hence contradiction by a2;
end;
