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 Th59:
  lim_filter( #Rseq,<. Frechet_Filter(NAT),Frechet_Filter(NAT).)) <> {}
  implies ex x being Real st
  lim_filter( #Rseq,<. Frechet_Filter(NAT),Frechet_Filter(NAT).)) = {x}
  proof
    assume
A1: lim_filter( #Rseq,<. Frechet_Filter(NAT),Frechet_Filter(NAT).)) <> {};
    reconsider s1 = Rseq as Function of [:NAT,NAT:],the carrier of R^1;
    consider x be object such that
A2: lim_filter( #Rseq,<. Frechet_Filter(NAT),Frechet_Filter(NAT).)) = {x}
      by A1,ZFMISC_1:131;
    x in lim_filter( #Rseq,<. Frechet_Filter(NAT),Frechet_Filter(NAT).))
      by A2,TARSKI:def 1;
    hence thesis by A2;
  end;
