theorem Th73:
  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(ProjMap1(f,x),Frechet_Filter(NAT)) <> {}) holds
  lim_filter(ProjMap1(f,t),Frechet_Filter(NAT)) = {lim ProjMap1(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(ProjMap1(f,x),Frechet_Filter(NAT)) <> {};
    lim_filter(ProjMap1(f,t),Frechet_Filter(NAT)) is non empty trivial by A2;
    then consider x be object such that
A3: lim_filter(ProjMap1(f,t),Frechet_Filter(NAT)) = {x} by ZFMISC_1:131;
    reconsider f1 = ProjMap1(f,t) as Function of NAT,R^1;
    reconsider seq1 = ProjMap1(seq,t) as Function of NAT,REAL;
    lim_f f1 = {lim seq1} by A1,A3,Th72;
    hence thesis;
  end;
