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 th54b:
Rseq1 is P-convergent & Rseq2 is P-convergent implies
 Rseq1-Rseq2 is P-convergent & P-lim(Rseq1-Rseq2) = P-lim Rseq1 - P-lim Rseq2
proof
   assume a1: Rseq1 is P-convergent & Rseq2 is P-convergent;
a2:now let e;
    assume a4: 0 < e; then
    consider N1 be Nat such that
a5:  for n,m st n>=N1 & m>=N1 holds
       |. Rseq1.(n,m) - P-lim Rseq1.| < e/2 by a1,def6;
    consider N2 be Nat such that
a6:  for n,m st n>=N2 & m>=N2 holds
       |. Rseq2.(n,m) - P-lim Rseq2.| < e/2 by a1,a4,def6;
    reconsider N=max(N1,N2) as Nat by TARSKI:1;
    take N;
    now let n,m;
     assume a8: n>=N & m>=N;
     N>=N1 & N>=N2 by XXREAL_0:25; then
     n>=N1 & n>=N2 & m>=N1 & m>=N2 by a8,XXREAL_0:2; then
a10: |. Rseq1.(n,m) - P-lim Rseq1.| < e/2 &
     |. Rseq2.(n,m) - P-lim Rseq2.| < e/2 by a5,a6; then
     |. P-lim Rseq2 - Rseq2.(n,m).| < e/2 by COMPLEX1:60; then
a9:  |. Rseq1.(n,m) - P-lim Rseq1.| + |.P-lim Rseq2 - Rseq2.(n,m) .| < e/2+e/2
       by a10,XREAL_1:8;
     (Rseq1-Rseq2).(n,m) = Rseq1.(n,m) - Rseq2.(n,m) by lmADD; then
     (Rseq1-Rseq2).(n,m) - (P-lim Rseq1 - P-lim Rseq2)
      = (Rseq1.(n,m) - P-lim Rseq1) + (P-lim Rseq2 - Rseq2.(n,m)); then
     |.(Rseq1-Rseq2).(n,m) - (P-lim Rseq1 - P-lim Rseq2).|
      <= |. Rseq1.(n,m) - P-lim Rseq1.|
       + |.P-lim Rseq2 - Rseq2.(n,m).| by COMPLEX1:56;
     hence |. (Rseq1-Rseq2).(n,m) - (P-lim Rseq1 - P-lim Rseq2).| < e
       by a9,XXREAL_0:2;
    end;
    hence for n,m be Nat st n>=N & m>=N holds
     |.(Rseq1-Rseq2).(n,m) - (P-lim Rseq1 - P-lim Rseq2).| < e;
   end;
   hence Rseq1-Rseq2 is P-convergent;
   hence thesis by a2,def6;
end;
