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 Th61:
  Rseq is P-convergent iff
    lim_filter( #Rseq,<. Frechet_Filter(NAT),Frechet_Filter(NAT).)) <> {}
  proof
    hereby
      assume Rseq is P-convergent;
      then consider x be Real such that
A1:   for e be Real st 0 < e ex N be Nat st
      for n,m be Nat st n >= N & m >= N holds |.Rseq.(n,m) - x.| < e;
      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) - x .| < 1/m
      proof
        let m be non zero Nat;
        0/m < 1/m by XREAL_1:74;
        hence thesis by A1;
      end;
      hence lim_filter( #Rseq,<. Frechet_Filter(NAT),Frechet_Filter(NAT).))
        <> {} by Th58;
    end;
    assume lim_filter( #Rseq,<. Frechet_Filter(NAT),Frechet_Filter(NAT).))
        <> {};
    then consider p be object such that
A2: p in lim_filter( #Rseq,<. Frechet_Filter(NAT),Frechet_Filter(NAT).))
      by XBOOLE_0:def 1;
    reconsider p as Real by A2;
    ex p0 be Real st for e be Real st 0 < e ex N be Nat st for n,m be Nat st
      n >= N & m >= N holds |.Rseq.(n,m) - p0.| < e
    proof
      take p;
      hereby
        let e be Real;
        assume 0 < e;
        then ex m st m is non zero & 1 / m < e by Th5;
        then consider m0 be non zero Nat such that
        m0 is non zero and
A3:     1 / m0 < e;
        consider n0 being Nat such that
A4:     for n1,n2 being Nat st n0 <= n1 & n0 <= n2 holds
        |. Rseq.(n1,n2) - p .| < 1 / m0 by Th58,A2;
        now
          take N = n0;
          hereby
            let n,m be Nat;
            assume n >= N & m >= N;
            then |.Rseq.(n,m) - p.| < 1/m0 by A4;
            hence |.Rseq.(n,m) - p.| < e by A3,XXREAL_0:2;
          end;
        end;
        hence ex N be Nat st for n,m be Nat st n>=N & m>=N holds
          |.Rseq.(n,m) - p.| < e;
      end;
    end;
    hence Rseq is P-convergent;
  end;
