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;

theorem Th65:
  lim_filter( #dblseq_ex_1,Frechet_Filter([:NAT,NAT:])) = {}
  proof
    assume lim_filter( #dblseq_ex_1,Frechet_Filter([:NAT,NAT:])) <> {}; then
    lim_filter( #dblseq_ex_1,Frechet_Filter([:NAT,NAT:])) is non empty; then
    consider x be object such that
A1: x in lim_filter( #dblseq_ex_1,Frechet_Filter([:NAT,NAT:]));
A2: lim_filter( #dblseq_ex_1,Frechet_Filter([:NAT,NAT:])) c=
      lim_filter( #dblseq_ex_1,<. all-square-uparrow.)) by Th56;
A3: <. all-square-uparrow .) = <. Frechet_Filter(NAT),Frechet_Filter(NAT).)
      by Th34,CARDFIL2:19,Th35;
    then consider y being Real such that
A4: lim_filter( #dblseq_ex_1,<. Frechet_Filter(NAT),Frechet_Filter(NAT).))
      = {y} by A1,A2,Th59;
    y = 0 by A4,Th64; then
A5: x = 0 by A1,A2,A3,A4,TARSKI:def 1;
    reconsider xp = 0 as Point of TopSpaceMetr(RealSpace) by Th64;
A6: Balls(xp) is basis of BOOL2F NeighborhoodSystem xp by CARDFIL3:6;
    consider yr be Point of RealSpace such that
A7: yr = xp and
A8: Balls(xp) = {Ball(yr,1/n) where n is Nat: n <> 0} by FRECHET:def 1;
    Ball(yr,1/2) in Balls(xp) by A8;
    then consider i,j such that
A9: for m,n st i <= m or j <= n holds dblseq_ex_1.(m,n) in Ball(yr,1/2)
      by A6,A5,A1,Th55;
    dblseq_ex_1.(0,j) in Ball(yr,1/2) by A9;
    then dblseq_ex_1.(0,j) in ].yr - 1/2,yr + 1/2.[ by FRECHET:7;
    then 1/(0+1) in ].yr - 1/2,yr + 1/2.[ by Def5;
    hence thesis by A7,XXREAL_1:4;
  end;
