
theorem
for f be Function of [:NAT,NAT:],ExtREAL, K be R_eal st
 K <> +infty & (for n,m be Nat holds f.(n,m) <= K) holds
   sup rng f < +infty
proof
   let f be Function of [:NAT,NAT:],ExtREAL, K be R_eal;
   assume A1: K <> +infty & (for n,m be Nat holds f.(n,m) <= K);
   now let x be ExtReal;
    assume x in rng f; then
    consider z be object such that
A2:  z in dom f & x = f.z by FUNCT_1:def 3;
    consider n,m be object such that
A3:  n in NAT & m in NAT & z = [n,m] by A2,ZFMISC_1:def 2;
    reconsider n,m as Element of NAT by A3;
    x = f.(n,m) by A2,A3;
    hence x <= K by A1;
   end; then
   K is UpperBound of rng f by XXREAL_2:def 1; then
   sup rng f <= K by XXREAL_2:def 3;
   hence thesis by A1,XXREAL_0:2,4;
end;
