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;
