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 Th74:
  for t being Element of NAT, f being Function of [:NAT,NAT:],R^1
  for seq being Function of [:NAT,NAT:],REAL st f = seq &
  (for x being Element of NAT holds
  lim_filter(ProjMap2(f,x),Frechet_Filter(NAT)) <> {}) holds
  lim_filter(ProjMap2(f,t),Frechet_Filter(NAT)) = {lim ProjMap2(seq,t)}
  proof
    let t be Element of NAT,
        f be Function of [:NAT,NAT:],R^1;
    let seq be Function of [:NAT,NAT:],REAL;
    assume that
A1: f = seq and
A2: for x being Element of NAT holds
      lim_filter(ProjMap2(f,x),Frechet_Filter(NAT)) <> {};
    lim_filter(ProjMap2(f,t),Frechet_Filter(NAT)) is non empty trivial by A2;
    then consider x be object such that
A3: lim_filter(ProjMap2(f,t),Frechet_Filter(NAT)) = {x} by ZFMISC_1:131;
    reconsider f1 = ProjMap2(f,t) as Function of NAT,R^1;
    reconsider seq1 = ProjMap2(seq,t) as Function of NAT,REAL;
    lim_f f1 = {lim seq1} by A1,A3,Th72;
    hence thesis;
  end;
