 reserve Rseq, Rseq1, Rseq2 for Function of [:NAT,NAT:],REAL;

theorem th105:
  Rseq is nonnegative-yielding implies
   ( for i1,i2,j be Nat st i1 <= i2 holds
      (Partial_Sums Rseq).(i1,j) <= (Partial_Sums Rseq).(i2,j) )
 & ( for i,j1,j2 be Nat st j1 <= j2 holds
      (Partial_Sums Rseq).(i,j1) <= (Partial_Sums Rseq).(i,j2) )
proof
   assume
A1: Rseq is nonnegative-yielding;
   hereby let i1,i2,j be Nat;
    assume A3: i1 <= i2;
    defpred P[Nat] means
      (Partial_Sums Rseq).(i1,$1) <= (Partial_Sums Rseq).(i2,$1);
    (Partial_Sums Rseq).(i1,0) = (Partial_Sums_in_cod1 Rseq).(i1,0)
  & (Partial_Sums Rseq).(i2,0) = (Partial_Sums_in_cod1 Rseq).(i2,0)
       by DefCS; then
A4: P[0] by A3,A1,th1003;
A5: for k be Nat st P[k] holds P[k+1]
    proof
     let k be Nat;
     assume A6: P[k];
A7:  (Partial_Sums_in_cod1 Rseq).(i1,k+1)
       <= (Partial_Sums_in_cod1 Rseq).(i2,k+1) by A3,A1,th1003;
     (Partial_Sums Rseq).(i1,k+1)
      = (Partial_Sums Rseq).(i1,k) + (Partial_Sums_in_cod1 Rseq).(i1,k+1)
   & (Partial_Sums Rseq).(i2,k+1)
      = (Partial_Sums Rseq).(i2,k) + (Partial_Sums_in_cod1 Rseq).(i2,k+1)
        by DefCS;
     hence thesis by A6,A7,XREAL_1:7;
    end;
    for k be Nat holds P[k] from NAT_1:sch 2(A4,A5);
    hence (Partial_Sums Rseq).(i1,j) <= (Partial_Sums Rseq).(i2,j);
   end;
   hereby let i,j1,j2 be Nat;
    assume B3: j1 <= j2;
    defpred Q[Nat] means
     (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).($1,j1)
       <= (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).($1,j2);
    (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).(0,j1)
      = (Partial_Sums_in_cod2 Rseq).(0,j1)
  & (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).(0,j2)
      = (Partial_Sums_in_cod2 Rseq).(0,j2) by DefRS; then
B4: Q[0] by B3,A1,th1003;
B5: for k be Nat st Q[k] holds Q[k+1]
    proof
     let k be Nat;
     assume C6: Q[k];
C7:  (Partial_Sums_in_cod2 Rseq).(k+1,j1)
      <= (Partial_Sums_in_cod2 Rseq).(k+1,j2) by B3,A1,th1003;
     (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).(k+1,j1)
      = (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).(k,j1)
       + (Partial_Sums_in_cod2 Rseq).(k+1,j1)
   & (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).(k+1,j2)
      = (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).(k,j2)
       + (Partial_Sums_in_cod2 Rseq).(k+1,j2) by DefRS;
     hence thesis by C6,C7,XREAL_1:7;
    end;
B6: for k be Nat holds Q[k] from NAT_1:sch 2(B4,B5);
    (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).(i,j1)
      = (Partial_Sums Rseq).(i,j1)
  & (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).(i,j2)
      = (Partial_Sums Rseq).(i,j2) by th103;
    hence (Partial_Sums Rseq).(i,j1) <= (Partial_Sums Rseq).(i,j2) by B6;
   end;
end;
