reserve            x for object,
               X,Y,Z for set,
         i,j,k,l,m,n for Nat,
                 r,s for Real,
                  no for Element of OrderedNAT,
                   A for Subset of [:NAT,NAT:];
reserve X,Y,X1,X2 for non empty set,
          cA1,cB1 for filter_base of X1,
          cA2,cB2 for filter_base of X2,
              cF1 for Filter of X1,
              cF2 for Filter of X2,
             cBa1 for basis of cF1,
             cBa2 for basis of cF2;
reserve T for non empty TopSpace,
        s for Function of [:NAT,NAT:], the carrier of T,
        M for Subset of the carrier of T;
reserve cF3,cF4 for Filter of the carrier of T;
reserve Rseq for Function of [:NAT,NAT:],REAL;
reserve f for Function of [#]OrderedNAT,R^1,
        seq for Function of NAT,REAL;

theorem
  for Y being non empty Hausdorff TopSpace,
      f being Function of [:NAT,NAT:],Y st
  (for x being Element of NAT holds
  lim_filter(ProjMap1(f,x),Frechet_Filter(NAT)) <> {}) & f = Rseq & Y = R^1
  holds
  lim_in_cod2(f,Frechet_Filter(NAT)) = lim_in_cod2 Rseq
  proof
    let Y be non empty Hausdorff TopSpace, f be Function of [:NAT,NAT:],Y;
    assume that
A1: for x being Element of NAT holds
      lim_filter(ProjMap1(f,x),Frechet_Filter(NAT)) <> {} and
A2: f = Rseq and
A3: Y = R^1;
    now
      dom lim_in_cod2(f,Frechet_Filter(NAT)) = NAT by FUNCT_2:def 1;
      hence dom lim_in_cod2(f,Frechet_Filter(NAT))
        = dom lim_in_cod2 Rseq by FUNCT_2:def 1;
      thus for t be object st
        t in dom lim_in_cod2(f,Frechet_Filter(NAT))
        holds (lim_in_cod2(f,Frechet_Filter(NAT))).t
          = (lim_in_cod2 Rseq).t
      proof
        let t be object;
        assume t in dom lim_in_cod2(f,Frechet_Filter(NAT));
        then reconsider t1 = t as Element of NAT;
A4:     {(lim_in_cod2(f,Frechet_Filter(NAT))).t1}
          = lim_filter(ProjMap1(f,t1),Frechet_Filter(NAT)) by A1,Def7;
        lim_filter(ProjMap1(f,t1),Frechet_Filter(NAT))
          = {lim ProjMap1(Rseq,t1)} by A1,A3,A2,Th73
         .= {(lim_in_cod2 Rseq).t1} by DBLSEQ_1:def 6;
        hence thesis by A4,ZFMISC_1:3;
      end;
    end;
    hence lim_in_cod2(f,Frechet_Filter(NAT)) = lim_in_cod2 Rseq
      by FUNCT_1:def 11;
  end;
