
theorem
  for f be Function of [:NAT,NAT:],ExtREAL, r be Real st
   (for n,m be Nat holds f.(n,m) = r) holds
    f is P-convergent_to_finite_number & P-lim f = r
proof
   let f be Function of [:NAT,NAT:],ExtREAL, r be Real;
   assume
A1: for n,m be Nat holds f.(n,m) = r;
A2:now
    reconsider N=1 as Nat;
    let p be Real;
    assume
A3:  0 < p;
    take N;
    let n,m be Nat such that n>=N & m>=N;
    f.(n,m) = r by A1;
    hence |. f.(n,m) - r .| < p by A3,XXREAL_3:7,EXTREAL1:16;
   end;
   hence
A4: f is P-convergent_to_finite_number;
   then f is P-convergent;
   hence thesis by A2,A4,Def5;
end;
