
theorem Th116:
for X,Y be set, A be SetSequence of X, B be SetSequence of Y,
 C be SetSequence of [:X,Y:] st A is non-descending & B is non-descending &
 (for n be Nat holds C.n = [:A.n,B.n:]) holds
  C is non-descending & C is convergent & Union C = [:Union A,Union B:]
proof
   let X,Y be set, A be SetSequence of X, B be SetSequence of Y,
   C be SetSequence of [:X,Y:];
   assume that
A1: A is non-descending and
A2: B is non-descending and
A3: (for n be Nat holds C.n = [:A.n,B.n:]);
   for n,m be Nat st n <= m holds C.n c= C.m
   proof
    let n,m be Nat;
    assume n <= m; then
    A.n c= A.m & B.n c= B.m by A1,A2,PROB_1:def 5; then
    [:A.n,B.n:] c= [:A.m,B.m:] by ZFMISC_1:96; then
    C.n c= [:A.m,B.m:] by A3;
    hence C.n c= C.m by A3;
   end;
   hence C is non-descending by PROB_1:def 5;
   hence C is convergent by SETLIM_1:63;
   now let z be set;
    assume z in [:Union A,Union B:]; then
    consider x,y be object such that
A6:  x in Union A & y in Union B & z = [x,y] by ZFMISC_1:def 2;
A7: x in union rng A & y in union rng B by A6,CARD_3:def 4; then
    consider A1 be set such that
A8:  x in A1 & A1 in rng A by TARSKI:def 4;
    consider n be object such that
A9:  n in dom A & A1 = A.n by A8,FUNCT_1:def 3;
    reconsider n as Nat by A9;
    consider B1 be set such that
A10: y in B1 & B1 in rng B by A7,TARSKI:def 4;
    consider m be object such that
A11: m in dom B & B1 = B.m by A10,FUNCT_1:def 3;
    reconsider m as Nat by A11;
    reconsider N = max(n,m) as Element of NAT by ORDINAL1:def 12;
    A.n c= A.N & B.m c= B.N by A1,A2,XXREAL_0:25,PROB_1:def 5; then
    z in [:A.N,B.N:] by A6,A8,A9,A10,A11,ZFMISC_1:def 2; then
    z in C.N & C.N in rng C by A3,FUNCT_2:112; then
    z in union rng C by TARSKI:def 4;
    hence z in Union C by CARD_3:def 4;
   end; then
A12: [:Union A,Union B:] c= Union C;
   now let z be set;
    assume z in Union C; then
    z in union rng C by CARD_3:def 4; then
    consider C1 be set such that
A13: z in C1 & C1 in rng C by TARSKI:def 4;
    consider n be object such that
A14: n in dom C & C1 = C.n by A13,FUNCT_1:def 3;
    reconsider n as Element of NAT by A14;
    z in [:A.n,B.n:] by A3,A13,A14; then
    consider x,y be object such that
A15: x in A.n & y in B.n & z = [x,y] by ZFMISC_1:def 2;
    A.n in rng A & B.n in rng B by FUNCT_2:112; then
    x in union rng A & y in union rng B by A15,TARSKI:def 4; then
    x in Union A & y in Union B by CARD_3:def 4;
    hence z in [:Union A,Union B:] by A15,ZFMISC_1:def 2;
   end; then
   Union C c= [:Union A,Union B:];
   hence Union C = [:Union A,Union B:] by A12;
end;
