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 Th60:
  Rseq is P-convergent implies P-lim Rseq in
    lim_filter( #Rseq,<. Frechet_Filter(NAT),Frechet_Filter(NAT).))
  proof
    assume
A1: Rseq is P-convergent;
    for m being non zero Nat ex n being Nat st for n1,n2 being Nat st
      n <= n1 & n <= n2 holds |. Rseq.(n1,n2) - P-lim Rseq .| < 1/m
    proof
      let m be non zero Nat;
      0/m < 1/m by XREAL_1:74;
      hence thesis by A1,DBLSEQ_1:def 2;
    end;
    hence P-lim Rseq in
    lim_filter( #Rseq,<. Frechet_Filter(NAT),Frechet_Filter(NAT).)) by Th58;
  end;
