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;
