
theorem
for f1 be without-infty Function of [:NAT,NAT:],ExtREAL,
    f2 be Function of [:NAT,NAT:],ExtREAL, c be Real
  st 0 <= c
  & ( for n1,m1,n2,m2 be Nat st n1 <= n2 & m1 <= m2 holds
        f1.(n1,m1) <= f1.(n2,m2) )
  & ( for n,m be Nat holds f2.(n,m) = c * f1.(n,m) )
 holds
  (for n1,m1,n2,m2 be Nat st n1 <= n2 & m1 <= m2 holds
      f2.(n1,m1) <= f2.(n2,m2) )
 & f2 is without-infty & f2 is P-convergent
 & P-lim f2 = sup rng f2 & P-lim f2 = c * P-lim f1
proof
   let f1 be without-infty Function of [:NAT,NAT:],ExtREAL,
    f2 be Function of [:NAT,NAT:],ExtREAL, c be Real;
   assume that
A1: 0 <= c and
A2: for n1,m1,n2,m2 be Nat st n1 <= n2 & m1 <= m2 holds
        f1.(n1,m1) <= f1.(n2,m2) and
A3: for n,m be Nat holds f2.(n,m) = c * f1.(n,m);
A5:sup rng f1 = P-lim f1 by A2,Th96;
   thus
A6: now let n1,m1,n2,m2 be Nat;
    assume n1 <= n2 & m1 <= m2; then
    c * f1.(n1,m1) <= c * f1.(n2,m2) by A1,A2,XXREAL_3:71; then
    f2.(n1,m1) <= c * f1.(n2,m2) by A3;
    hence f2.(n1,m1) <= f2.(n2,m2) by A3;
   end;
   thus f2 is without-infty by A1,A3,Th104;
   thus f2 is P-convergent & P-lim f2 = sup rng f2 by A6,Th96;
   sup rng f2 = P-lim f2 by A6,Th96;
   hence thesis by A1,A3,A5,Th104;
end;
