
theorem Th85:
for f be Function of [:NAT,NAT:],ExtREAL, seq be ExtREAL_sequence, n,m be Nat
 holds
  ( (for i,j be Nat holds f.(i,j) <= seq.i) implies
            (Partial_Sums_in_cod1 f).(n,m) <= (Partial_Sums seq).n )
& ( (for i,j be Nat holds f.(i,j) <= seq.j) implies
            (Partial_Sums_in_cod2 f).(n,m) <= (Partial_Sums seq).m )
proof
   let f be Function of [:NAT,NAT:],ExtREAL, seq be ExtREAL_sequence,
       n,m be Nat;
   hereby assume
A1: for i,j be Nat holds f.(i,j) <= seq.i;
    defpred P[Nat] means
     (Partial_Sums_in_cod1 f).($1,m) <= (Partial_Sums seq).$1;
A2: (Partial_Sums_in_cod1 f).(0,m) = f.(0,m) by DefRSM;
    (Partial_Sums seq).0 = seq.0 by MESFUNC9:def 1; then
A3: P[0] by A1,A2;
A4: for k be Nat st P[k] holds P[k+1]
    proof
     let k be Nat;
     assume A5: P[k];
A6:  f.(k+1,m) <= seq.(k+1) by A1;
A7:  (Partial_Sums_in_cod1 f).(k+1,m)
      = (Partial_Sums_in_cod1 f).(k,m) + f.(k+1,m) by DefRSM;
     (Partial_Sums seq).(k+1) = (Partial_Sums seq).k + seq.(k+1)
       by MESFUNC9:def 1;
     hence thesis by A5,A6,A7,XXREAL_3:36;
    end;
    for k be Nat holds P[k] from NAT_1:sch 2(A3,A4);
    hence (Partial_Sums_in_cod1 f).(n,m) <= (Partial_Sums seq).n;
   end;
   assume
A1: for i,j be Nat holds f.(i,j) <= seq.j;
   defpred P[Nat] means
    (Partial_Sums_in_cod2 f).(n,$1) <= (Partial_Sums seq).$1;
A2:(Partial_Sums_in_cod2 f).(n,0) = f.(n,0) by DefCSM;
   (Partial_Sums seq).0 = seq.0 by MESFUNC9:def 1; then
A3:P[0] by A1,A2;
A4:for k be Nat st P[k] holds P[k+1]
   proof
    let k be Nat;
    assume A5: P[k];
A6: f.(n,k+1) <= seq.(k+1) by A1;
A7: (Partial_Sums_in_cod2 f).(n,k+1)
     = (Partial_Sums_in_cod2 f).(n,k) + f.(n,k+1) by DefCSM;
    (Partial_Sums seq).(k+1) = (Partial_Sums seq).k + seq.(k+1)
      by MESFUNC9:def 1;
    hence thesis by A5,A6,A7,XXREAL_3:36;
   end;
   for k be Nat holds P[k] from NAT_1:sch 2(A3,A4);
   hence thesis;
end;
