
theorem
for f be nonnegative Function of [:NAT,NAT:],ExtREAL,
    seq be ExtREAL_sequence st
 (for m be Element of NAT holds seq.m = lim_inf ProjMap1(f,m))
  holds Sum seq <= lim_inf lim_in_cod1(Partial_Sums_in_cod1 f)
proof
   let f be nonnegative Function of [:NAT,NAT:],ExtREAL,
       seq be ExtREAL_sequence;
   assume A1: for m be Element of NAT holds seq.m = lim_inf ProjMap1(f,m);
   now let m be Element of NAT;
    ProjMap1(f,m) = ProjMap2(~f,m) by Th32;
    hence seq.m = lim_inf ProjMap2(~f,m) by A1;
   end; then
A2:Sum seq <= lim_inf lim_in_cod2(Partial_Sums_in_cod2 ~f) by Th83;
   lim_in_cod2(Partial_Sums_in_cod2 ~f)
    = lim_in_cod1(~Partial_Sums_in_cod2 ~f) by Th38
   .= lim_in_cod1(Partial_Sums_in_cod1 ~(~f)) by Th40;
   hence Sum seq <= lim_inf lim_in_cod1(Partial_Sums_in_cod1 f)
     by A2,DBLSEQ_2:7;
end;
