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;

theorem
for r be Element of REAL holds P-lim ([:NAT,NAT:] --> r) = r
proof
   let r be Element of REAL;
   set Rseq = [:NAT,NAT:] --> r;
a1:for n,m be Nat holds Rseq.(n,m) = r
   proof
    let n,m be Nat;
    n is Element of NAT & m is Element of NAT by ORDINAL1:def 12;
    hence Rseq.(n,m) = r by FUNCOP_1:70;
   end;
   now let e be Real;
    assume a2: 0<e;
a4: now let n,m such that n>=0 & m>=0;
     Rseq.(n,m) = r by a1;
     hence |. Rseq.(n,m) - r.| < e by a2,COMPLEX1:44;
    end;
    reconsider N = 0 as Nat;
    take N;
    thus for n,m st n>=N & m>=N holds |. Rseq.(n,m) - r.| < e by a4;
   end;
   hence P-lim ([:NAT,NAT:] --> r) = r by def6;
end;
