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;
reserve f for Function of [#]OrderedNAT,R^1,
        seq for Function of NAT,REAL;

theorem Th71:
  f = seq & lim_f f <> {} implies lim_f f = {lim seq}
  proof
    assume that
A1: f = seq and
A2: lim_f f <> {};
    consider x be object such that
A3: x in lim_f f by A2, XBOOLE_0:def 1;
    reconsider y = x as Point of R^1 by A3;
    consider u be object such that
A4: lim_f f = {u} by A3,ZFMISC_1:131;
    lim_f f = {lim seq}
    proof
      lim_f f c= {lim seq}
      proof
        let t be object;
        assume
A5:     t in lim_f f;
        then reconsider t1 = t as Real;
A6:     seq is convergent &
        ex z being Real st z in lim_f f & for p being Real st 0 < p holds
          ex n be Nat st for m be Nat st n <= m holds |.seq.m - z.| < p
          by A1,A2,Th70;
        consider z being Real such that
A7:     z in lim_f f and
A8:     for p being Real st 0 < p holds ex n be Nat st for m be Nat st
          n <= m holds |.seq.m - z.| < p by A1,A2,Th70;
        t = u & z = u by A5,A7,A4,TARSKI:def 1;
        then t1 = lim seq by A8,A6,SEQ_2:def 7;
        hence thesis by TARSKI:def 1;
      end;
      hence thesis by A4,ZFMISC_1:3;
    end;
    hence thesis;
  end;
