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;
reserve Y for non empty TopSpace,
        x for Point of Y,
        f for Function of [:X1,X2:],Y;

theorem Th80:
  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 X1 holds lim_filter(ProjMap1(f,x),cF2) <> {})
  holds lim_filter(f,<.cF1,cF2.)) = lim_filter(lim_in_cod2(f,cF2),cF1)
  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 X1 holds lim_filter(ProjMap1(f,x),cF2) <> {};
    consider y be object such that
A3: lim_filter(f,<.cF1,cF2.)) = {y} by A1,ZFMISC_1:131;
A4: y in lim_filter(f,<.cF1,cF2.)) by A3,TARSKI:def 1;
A5: lim_filter(f,<.cF1,cF2.)) c= lim_filter(lim_in_cod2(f,cF2),cF1)
      by A2,Th79;
    lim_filter(lim_in_cod2(f,cF2),cF1) is non empty trivial by A3,A5;
    then ex z be object st lim_filter(lim_in_cod2(f,cF2),cF1) = {z}
      by ZFMISC_1:131;
    hence thesis by A3,A5,A4,TARSKI:def 1;
  end;
