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 x being Element of NAT holds
    lim_filter(ProjMap1( #Rseq ,x),Frechet_Filter(NAT)) <> {})
  iff
  Rseq is convergent_in_cod2
  proof
    hereby
      assume
A1:   for x being Element of NAT holds
      lim_filter(ProjMap1( #Rseq ,x),Frechet_Filter(NAT)) <> {};
      now
        let m be Element of NAT;
        lim_filter(ProjMap1( #Rseq ,m),Frechet_Filter(NAT)) is non empty by A1;
        then consider r be Element of TopSpaceMetr(RealSpace) such that
A2:     r in lim_filter(ProjMap1( #Rseq ,m),Frechet_Filter(NAT));
A3:     r in lim_f ProjMap1( #Rseq ,m) by A2;
A4:     Balls(r) is basis of BOOL2F NeighborhoodSystem r by CARDFIL3:6;
        reconsider r1 = r as Real;
        now
          let e be Real;
          assume 0 < e;
          then consider m1 be Nat such that
A5:       m1 is non zero and
A6:       1 / m1 < e by Th5;
          consider y be Point of RealSpace such that
A7:       y = r and
A8:       Balls(r) = { Ball(y,1/n) where n is Nat: n <> 0 } by FRECHET:def 1;
A9:       Ball(y,1/m1) c= Ball(y,e) by A6,PCOMPS_1:1;
          Ball(y,1/m1) in Balls(r) by A8,A5;
          then consider i be Nat such that
A10:      for j be Nat st i <=j holds (ProjMap1( #Rseq,m)).j in Ball(y,1/m1)
            by A4,A3,CARDFIL2:97;
          thus ex N be Nat st for n be Nat st N <= n holds
            |.ProjMap1(Rseq,m).n - r1.| < e
          proof
            take i;
            let j be Nat;
            assume i <= j;
            then (ProjMap1( #Rseq,m)).j in Ball(y,e) by A9,A10;
            then (ProjMap1( #Rseq,m)).j in ]. y - e,y + e .[ by FRECHET:7;
            hence thesis by A7,FCONT_3:1;
          end;
        end;
        hence ProjMap1(Rseq,m) is convergent by SEQ_2:def 6;
      end;
      hence Rseq is convergent_in_cod2;
    end;
    assume
A11: Rseq is convergent_in_cod2;
    now
      let x be Element of NAT;
      consider r be Real such that
A12:  for p be Real st 0 < p ex n st for m be Nat st n <= m holds
        |.ProjMap1(Rseq,x).m - r.| < p by A11,SEQ_2:def 6;
      reconsider r1 = r as Point of TopSpaceMetr(RealSpace) by XREAL_0:def 1;
A13:  Balls(r1) is basis of BOOL2F NeighborhoodSystem r1 by CARDFIL3:6;
      for b be Element of Balls(r1) holds ex i be Nat st for j be Nat st
        i <= j holds ProjMap1( #Rseq,x).j in b
      proof
        let b be Element of Balls(r1);
        consider y be Point of RealSpace such that
A14:    y = r1 and
A15:    Balls(r1) = { Ball(y,1/n) where n is Nat: n <> 0 } by FRECHET:def 1;
        b in { Ball(y,1/n) where n is Nat: n <> 0 } by A15;
        then consider n0 be Nat such that
A16:    b = Ball(y,1/n0) and
A17:    n0 <> 0;
        0 < n0 & 0 * n0 < 1 by A17;
        then consider n1 be Nat such that
A18:    for m be Nat st n1 <= m holds |.ProjMap1(Rseq,x).m - r.| < 1/n0
          by A12,XREAL_1:81;
        now
          take n1;
          hereby
            let j be Nat;
            assume n1 <= j; then
A19:        |.ProjMap1(Rseq,x).j - r.| < 1/n0 by A18;
            ProjMap1(Rseq,x).j = r + (ProjMap1(Rseq,x).j - r);
            then ProjMap1(Rseq,x).j in ]. r - 1/n0, r + 1/n0.[
              by A19,FCONT_3:3;
            hence ProjMap1( #Rseq,x).j in b by A14,A16,FRECHET:7;
          end;
        end;
        hence ex i be Nat st for j be Nat st i <= j holds
          ProjMap1( #Rseq,x).j in b;
      end;
      then lim_f ProjMap1( #Rseq ,x) is non empty by A13,CARDFIL2:97;
      hence lim_filter(ProjMap1( #Rseq ,x),Frechet_Filter(NAT)) is non empty;
    end;
    hence thesis;
  end;
