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

theorem
  Rseq is non-decreasing & Rseq is convergent_in_cod1 implies
   lim_in_cod1 Rseq is non-decreasing
proof
   assume that
a1: Rseq is non-decreasing and
a2: Rseq is convergent_in_cod1;
   now let i,j be Nat;
    assume a4: i <= j;
    reconsider n1=i, n2=j as Element of NAT by ORDINAL1:def 12;
a5: ProjMap2(Rseq,n1) is convergent & ProjMap2(Rseq,n2) is convergent by a2;
    now let m be Nat;
     m in NAT by ORDINAL1:def 12; then
     ProjMap2(Rseq,n1).m = Rseq.(m,n1) &
     ProjMap2(Rseq,n2).m = Rseq.(m,n2) by MESFUNC9:def 7;
     hence ProjMap2(Rseq,n1).m <= ProjMap2(Rseq,n2).m by a1,a4;
    end; then
    lim ProjMap2(Rseq,n1) <= lim ProjMap2(Rseq,n2) by a5,SEQ_2:18; then
    (lim_in_cod1 Rseq).n1 <= lim ProjMap2(Rseq,n2) by DBLSEQ_1:def 5;
    hence (lim_in_cod1 Rseq).i <= (lim_in_cod1 Rseq).j by DBLSEQ_1:def 5;
   end;
   hence lim_in_cod1 Rseq is non-decreasing by SEQM_3:6;
end;
