
theorem
for X1,X2,A,B be set, F1 be SetSequence of X1, F2 be SetSequence of X2,
F be SetSequence of [:X1,X2:] st
  F1 is non-descending & lim F1 = A & F2 is non-descending & lim F2 = B &
  (for n be Nat holds F.n = [:F1.n,F2.n:])
holds lim F = [:A,B:]
proof
   let X1,X2,A,B be set, F1 be SetSequence of X1, F2 be SetSequence of X2,
   F be SetSequence of [:X1,X2:];
   assume that
A1: F1 is non-descending and
A2: lim F1 = A and
A3: F2 is non-descending and
A4: lim F2 = B and
A5: for n be Nat holds F.n = [:F1.n,F2.n:];
   now let n be Nat;
    F.n = [:F1.n,F2.n:] & F.(n+1) = [:F1.(n+1),F2.(n+1):] by A5;
    hence F.n c= F.(n+1) by A1,A3,Th4;
   end; then
   F is non-descending by PROB_2:7; then
A6:lim F = Union F by SETLIM_1:63;
   Union F1 = A & Union F2 = B by A1,A2,A3,A4,SETLIM_1:63; then
A8:union rng F1 = A & union rng F2 = B by CARD_3:def 4; then
A7: [:A,B:]
    = union { [:P,Q:] where P is Element of rng F1, Q is Element of rng F2 :
               P in rng F1 & Q in rng F2} by LATTICE5:2;
   now let z be object;
    assume z in [:A,B:]; then
    consider Z be set such that
X1:  z in Z
   & Z in { [:A,B:] where A is Element of rng F1, B is Element of rng F2 :
               A in rng F1 & B in rng F2} by A7,TARSKI:def 4;
    consider A be Element of rng F1, B be Element of rng F2 such that
X2:  Z = [:A,B:] & A in rng F1 & B in rng F2 by X1;
    consider n1 be Element of NAT such that
X3:  n1 in dom F1 & A = F1.n1 by PARTFUN1:3;
    consider n2 be Element of NAT such that
X4:  n2 in dom F2 & B = F2.n2 by PARTFUN1:3;
    set n = max(n1,n2);
    A c= F1.n & B c= F2.n by A1,A3,X3,X4,PROB_1:def 5,XXREAL_0:25; then
X5: Z c= [:F1.n,F2.n:] by X2,ZFMISC_1:96;
    n in NAT; then
    n in dom F by FUNCT_2:def 1; then
    F.n in rng F by FUNCT_1:3; then
    [:F1.n,F2.n:] in rng F by A5;
    hence z in union rng F by X1,X5,TARSKI:def 4;
   end; then
X6: [:A,B:] c= union rng F;
   now let z be object;
    assume z in union rng F; then
    consider Z be set such that
Y1:  z in Z & Z in rng F by TARSKI:def 4;
    consider n be Element of NAT such that
Y2:  n in dom F & Z = F.n by Y1,PARTFUN1:3;
Y3: Z = [:F1.n,F2.n:] by A5,Y2;
    dom F1 = NAT & dom F2 = NAT by FUNCT_2:def 1; then
    F1.n c= union rng F1 & F2.n c= union rng F2 by FUNCT_1:3,ZFMISC_1:74; then
    Z c= [:A,B:] by A8,Y3,ZFMISC_1:96;
    hence z in [:A,B:] by Y1;
   end; then
   union rng F c= [:A,B:];
   hence lim F = [:A,B:] by A6,X6,CARD_3:def 4;
end;
