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 & P-lim Rseq1 = P-lim Rseq2 &
(for n,m be Nat holds
   Rseq1.(n,m) <= Rseq.(n,m) & Rseq.(n,m) <= Rseq2.(n,m) )
 implies Rseq is P-convergent & P-lim Rseq = P-lim Rseq1
proof
   assume that
a1: Rseq1 is P-convergent & Rseq2 is P-convergent and
a2: P-lim Rseq1 = P-lim Rseq2 and
a3: for n,m be Nat holds
     Rseq1.(n,m) <= Rseq.(n,m) & Rseq.(n,m) <= Rseq2.(n,m);
a4:for e st 0<e
    ex N st for n,m st n>=N & m>=N holds |. Rseq.(n,m) - P-lim Rseq1.| < e
   proof
    let e;
    assume a5: 0<e; then
    consider N1 be Nat such that
a6:  for n,m st n>=N1 & m>=N1 holds |. Rseq1.(n,m) - P-lim Rseq1.| < e
       by a1,def6;
    consider N2 be Nat such that
a7:  for n,m st n>=N2 & m>=N2 holds |. Rseq2.(n,m) - P-lim Rseq1.| < e
       by a1,a2,a5,def6;
    reconsider N = max(N1,N2) as Nat by TARSKI:1;
    take N;
a8: max(N1,N2) >= N1 & max(N1,N2) >= N2 by XXREAL_0:25;
    hereby let n,m;
     assume n>=N & m>=N; then
     n>=N1 & m>=N1 & n>=N2 & m>=N2 by a8,XXREAL_0:2; then
a9:  |. Rseq1.(n,m) - P-lim Rseq1.| < e & |. Rseq2.(n,m) - P-lim Rseq1.| < e
       by a6,a7; then
a10: -e < - |. Rseq1.(n,m) - P-lim Rseq1.| by XREAL_1:24;
     - |. Rseq1.(n,m) - P-lim Rseq1.| <= Rseq1.(n,m) - P-lim Rseq1
       by ABSVALUE:4; then
a11: -e < Rseq1.(n,m) - P-lim Rseq1 by a10,XXREAL_0:2;
     Rseq2.(n,m) - P-lim Rseq1 <= |. Rseq2.(n,m) - P-lim Rseq1.|
       by ABSVALUE:4; then
a12: Rseq2.(n,m) - P-lim Rseq1 < e by a9,XXREAL_0:2;
a13: Rseq1.(n,m) - P-lim Rseq1 <= Rseq.(n,m) - P-lim Rseq1 &
     Rseq.(n,m) - P-lim Rseq1 <= Rseq2.(n,m) - P-lim Rseq1
       by a3,XREAL_1:9; then
     -e < Rseq.(n,m) - P-lim Rseq1 & Rseq.(n,m) - P-lim Rseq1 < e
       by a11,a12,XXREAL_0:2; then
a14: |. Rseq.(n,m) - P-lim Rseq1.| <= e by ABSVALUE:5;
     -(Rseq.(n,m) - P-lim Rseq1) <> e by a3,a11,XREAL_1:9; then
     |. Rseq.(n,m) - P-lim Rseq1.| <> e by a13,a12,ABSVALUE:1;
     hence |. Rseq.(n,m) - P-lim Rseq1.| < e by a14,XXREAL_0:1;
    end;
   end;
   hence Rseq is P-convergent;
   hence P-lim Rseq = P-lim Rseq1 by a4,def6;
end;
