theorem Th81:
  for X1,X2 being non empty set,
      cF1 being Filter of X1,
      cF2 being Filter of X2,
      Y being Hausdorff regular non empty TopSpace,
      f being Function of [:X1,X2:],Y st
  lim_filter(f,<.cF1,cF2.)) <> {} &
  (for x being Element of X2 holds lim_filter(ProjMap2(f,x),cF1) <> {})
  holds lim_filter(f,<.cF1,cF2.)) = lim_filter(lim_in_cod1(f,cF1),cF2)
  proof
    let X1,X2 be non empty set,
    cF1 be Filter of X1,
    cF2 be Filter of X2,
    Y be Hausdorff regular non empty TopSpace,
    f be Function of [:X1,X2:],Y;
    assume that
A1: lim_filter(f,<.cF1,cF2.)) <> {} and
A2: for x being Element of X2 holds lim_filter(ProjMap2(f,x),cF1) <> {};
    consider y be object such that
A3: lim_filter(f,<.cF1,cF2.)) = {y} by A1,ZFMISC_1:131;
A4: lim_filter(f,<.cF1,cF2.)) c= lim_filter(lim_in_cod1(f,cF1),cF2)
      by A2,Th78;
A5: y in lim_filter(f,<.cF1,cF2.)) by A3,TARSKI:def 1;
    lim_filter(lim_in_cod1(f,cF1),cF2) is non empty trivial by A4,A3;
    then ex z be object st lim_filter(lim_in_cod1(f,cF1),cF2) = {z}
      by ZFMISC_1:131;
    hence thesis by A3,A5,A4,TARSKI:def 1;
  end;
