
theorem
  for f be Function of [:NAT,NAT:],ExtREAL holds
  for n,m be Nat holds f.(n,m) <= sup rng f
proof
   let f be Function of [:NAT,NAT:],ExtREAL;
   hereby let n,m be Nat;
A0: n in NAT & m in NAT by ORDINAL1:def 12;
    dom f = [:NAT,NAT:] by FUNCT_2:def 1; then
    [n,m] in dom f by A0,ZFMISC_1:87; then
A1: f.(n,m) in rng f by FUNCT_1:def 3;
    sup rng f is UpperBound of rng f by XXREAL_2:def 3;
    hence f.(n,m) <= sup rng f by A1,XXREAL_2:def 1;
   end;
end;
