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

theorem
  Rseq is nonnegative-yielding implies
   (for m,n being Nat holds
     Rseq.(m,n) <= (Partial_Sums_in_cod1 Rseq).(m,n) &
     Rseq.(m,n) <= (Partial_Sums_in_cod2 Rseq).(m,n))
proof
  assume a1: Rseq is nonnegative-yielding;
  hereby let m1,n1 be Nat;
    reconsider m=m1, n=n1 as Element of NAT by ORDINAL1:def 12;
    now let j be Nat;
     j in NAT by ORDINAL1:def 12; then
     (ProjMap2(Rseq,n)).j = Rseq.(j,n) by MESFUNC9:def 7;
     hence (ProjMap2(Rseq,n)).j >= 0 by a1;
    end; then
    (ProjMap2(Rseq,n)).m <= Partial_Sums(ProjMap2(Rseq,n)).m
      by th101,RINFSUP1:def 3; then
    (ProjMap2(Rseq,n)).m <= (Partial_Sums_in_cod1 Rseq).(m,n) by th100;
    hence Rseq.(m1,n1) <= (Partial_Sums_in_cod1 Rseq).(m1,n1)
      by MESFUNC9:def 7;
    now let j be Nat;
     j in NAT by ORDINAL1:def 12; then
     (ProjMap1(Rseq,m)).j = Rseq.(m,j) by MESFUNC9:def 6;
     hence (ProjMap1(Rseq,m)).j >= 0 by a1;
    end; then
    ProjMap1(Rseq,m) is nonnegative-yielding; then
    (ProjMap1(Rseq,m)).n <= Partial_Sums(ProjMap1(Rseq,m)).n by th101; then
    (ProjMap1(Rseq,m)).n <= (Partial_Sums_in_cod2 Rseq).(m,n) by th100;
    hence Rseq.(m1,n1) <= (Partial_Sums_in_cod2 Rseq).(m1,n1)
      by MESFUNC9:def 6;
   end;
end;
