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 Th72:
  for f being Function of NAT,R^1 st f = seq & lim_f f <> {} holds
  lim_f f = {lim seq}
  proof
    let f be Function of NAT,R^1;
    assume that
A1: f = seq and
A2: lim_f f <> {};
    [#]OrderedNAT = NAT & the carrier of R^1 = REAL by STRUCT_0:def 3;
    then reconsider f1 = f as Function of [#]OrderedNAT,R^1;
    lim_f f = lim_f f1 by CARDFIL2:54;
    hence thesis by A1,A2,Th71;
  end;
