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
Rseq1 is P-convergent & Rseq2 is P-convergent &
(for n,m be Nat holds Rseq1.(n,m) <= Rseq2.(n,m)) implies
  P-lim Rseq1 <= P-lim Rseq2
proof
   assume a1: Rseq1 is P-convergent & Rseq2 is P-convergent;
   assume a2: for n,m be Nat holds Rseq1.(n,m) <= Rseq2.(n,m);
a3:Rseq2 - Rseq1 is P-convergent &
   P-lim(Rseq2-Rseq1) = P-lim Rseq2 - P-lim Rseq1 by a1,th54b;
   now let n,m;
    (Rseq2-Rseq1).(n,m) = Rseq2.(n,m) - Rseq1.(n,m) by lmADD;
    hence (Rseq2-Rseq1).(n,m) >= 0 by a2,XREAL_1:48;
   end;
   hence P-lim Rseq1 <= P-lim Rseq2 by a3,th55b,XREAL_1:49;
end;
