
theorem Th65:
for f be nonnegative Function of [:NAT,NAT:],ExtREAL holds
  ( f is convergent_in_cod1 implies lim_in_cod1 f is nonnegative )
& ( f is convergent_in_cod2 implies lim_in_cod2 f is nonnegative )
proof
   let f be nonnegative Function of [:NAT,NAT:],ExtREAL;
   hereby assume A2:f is convergent_in_cod1;
    now let n be object;
     assume n in dom(lim_in_cod1 f); then
     reconsider n1=n as Element of NAT;
A4:  (lim_in_cod1 f).n = lim ProjMap2(f,n1) by D1DEF5;
     for k be Nat holds 0 <= ProjMap2(f,n1).k by SUPINF_2:51;
     hence (lim_in_cod1 f).n >= 0 by A2,A4,MESFUNC9:10;
    end;
    hence lim_in_cod1 f is nonnegative by SUPINF_2:52;
   end;
   assume A2:f is convergent_in_cod2;
   now let n be object;
    assume n in dom(lim_in_cod2 f); then
    reconsider n1=n as Element of NAT;
A4: (lim_in_cod2 f).n = lim ProjMap1(f,n1) by D1DEF6;

    for k be Nat holds 0 <= ProjMap1(f,n1).k by SUPINF_2:51;
    hence (lim_in_cod2 f).n >= 0 by A2,A4,MESFUNC9:10;
   end;
   hence thesis by SUPINF_2:52;
end;
