
theorem
for f be nonnegative Function of [:NAT,NAT:],ExtREAL,
    seq be ExtREAL_sequence st
 (for m be Element of NAT holds
    ProjMap1(f,m) is non-decreasing & seq.m = lim ProjMap1(f,m))
 holds
   lim_in_cod1(Partial_Sums_in_cod1 f) is non-decreasing
 & Sum seq = lim 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
      ProjMap1(f,m) is non-decreasing & seq.m = lim ProjMap1(f,m);
   for m be Element of NAT holds
    ProjMap2(~f,m) is non-decreasing & seq.m = lim ProjMap2(~f,m)
   proof
    let m be Element of NAT;
    ProjMap1(f,m) is non-decreasing by A1;
    hence ProjMap2(~f,m) is non-decreasing by Th32;
    seq.m = lim ProjMap1(f,m) by A1;
    hence seq.m = lim ProjMap2(~f,m) by Th32;
   end; then
A2:lim_in_cod2(Partial_Sums_in_cod2 ~f) is non-decreasing
 & Sum seq = lim lim_in_cod2(Partial_Sums_in_cod2 ~f) by Th91;
   for n be Element of NAT holds
    (lim_in_cod2(Partial_Sums_in_cod2 ~f)).n
     = (lim_in_cod1(Partial_Sums_in_cod1 f)).n
   proof
    let n be Element of NAT;
    (lim_in_cod1(Partial_Sums_in_cod1 f)).n
     = lim ProjMap2(Partial_Sums_in_cod1 f,n) by D1DEF5
    .= lim ProjMap1(~Partial_Sums_in_cod1 f,n) by Th33
    .= lim ProjMap1(Partial_Sums_in_cod2 ~f,n) by Th40;
    hence thesis by D1DEF6;
   end;
   hence thesis by A2,FUNCT_2:def 8;
end;
