
theorem Th76:
for f be nonnegative Function of [:NAT,NAT:],ExtREAL, n,m be Nat holds
  (Partial_Sums_in_cod1 f).(n,m) >= f.(n,m)
& (Partial_Sums_in_cod2 f).(n,m) >= f.(n,m)
proof
   let f be nonnegative Function of [:NAT,NAT:],ExtREAL, n,m be Nat;
   defpred P[Nat] means
    $1 <= n implies (Partial_Sums_in_cod1 f).($1,m) >= f.($1,m);
A2:P[0] by DefRSM;
A5:for k be Nat st P[k] holds P[k+1]
   proof
    let k be Nat such that P[k];
    assume k+1 <= n;
    (Partial_Sums_in_cod1 f).(k+1,m)
     = (Partial_Sums_in_cod1 f).(k,m) + f.(k+1,m) by DefRSM;
    hence (Partial_Sums_in_cod1 f).(k+1,m) >= f.(k+1,m)
      by SUPINF_2:51,XXREAL_3:39;
   end;
   for k be Nat holds P[k] from NAT_1:sch 2(A2,A5);
   hence (Partial_Sums_in_cod1 f).(n,m) >= f.(n,m);
   defpred Q[Nat] means
    $1 <= m implies (Partial_Sums_in_cod2 f).(n,$1) >= f.(n,$1);
A2:Q[0] by DefCSM;
A5:for k be Nat st Q[k] holds Q[k+1]
   proof
    let k be Nat such that Q[k];
    assume k+1 <= m;
    (Partial_Sums_in_cod2 f).(n,k+1)
     = (Partial_Sums_in_cod2 f).(n,k) + f.(n,k+1) by DefCSM;
    hence (Partial_Sums_in_cod2 f).(n,k+1) >= f.(n,k+1)
      by SUPINF_2:51,XXREAL_3:39;
   end;
   for k be Nat holds Q[k] from NAT_1:sch 2(A2,A5);
   hence (Partial_Sums_in_cod2 f).(n,m) >= f.(n,m);
end;
