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

theorem
  Rseq is non-decreasing & Rseq is convergent_in_cod2 implies
   lim_in_cod2 Rseq is non-decreasing
proof
   assume that
a1: Rseq is non-decreasing and
a2: Rseq is convergent_in_cod2;
   now let i,j be Nat;
    assume a4: i<=j;
    reconsider m1=i, m2=j as Element of NAT by ORDINAL1:def 12;
p1: ProjMap1(Rseq,m1) is convergent & ProjMap1(Rseq,m2) is convergent by a2;
    now let n be Nat;
     n in NAT by ORDINAL1:def 12; then
     ProjMap1(Rseq,m1).n = Rseq.(m1,n) &
     ProjMap1(Rseq,m2).n = Rseq.(m2,n) by MESFUNC9:def 6;
     hence ProjMap1(Rseq,m1).n <= ProjMap1(Rseq,m2).n by a1,a4;
    end; then
    lim ProjMap1(Rseq,m1) <= lim ProjMap1(Rseq,m2) by p1,SEQ_2:18; then
    (lim_in_cod2 Rseq).m1 <= lim ProjMap1(Rseq,m2) by DBLSEQ_1:def 6;
    hence (lim_in_cod2 Rseq).i <= (lim_in_cod2 Rseq).j by DBLSEQ_1:def 6;
   end;
   hence lim_in_cod2 Rseq is non-decreasing by SEQM_3:6;
end;
