
theorem
for X be set, S be with_empty_element semi-diff-closed cap-closed
   Subset-Family of X,
    F be disjoint_valued FinSequence of S,
    R be non empty preBoolean Subset-Family of X
 st S c= R & Union F in R
 holds for i be Nat holds Union (F|i) in R
proof
   let X be set,
       S be with_empty_element semi-diff-closed cap-closed Subset-Family of X,
       F be disjoint_valued FinSequence of S,
       R be non empty preBoolean Subset-Family of X;
   assume A1: S c= R & Union F in R;

   defpred P[Nat] means Union (F|$1) in R;

   union rng (F|0) = {} by ZFMISC_1:2; then
   Union (F|0) = {} by CARD_3:def 4; then
A3:P[0] by FINSUB_1:7;

A4:for i be Nat st P[i] holds P[i+1]
   proof
    let i be Nat;
    assume A5: P[i];
    per cases;
    suppose i >= len F; then
     F|i = F & F|(i+1) = F by NAT_1:12,FINSEQ_1:58;
     hence P[i+1] by A5;
    end;
    suppose i < len F; then
A8:  i+1 <= len F by NAT_1:13;

     set F1 = F|(i+1);

A9:  F1|i = F|i by NAT_1:12,FINSEQ_1:82;

     F1 = F1|i ^ <*F1.(i+1)*> by A8,FINSEQ_1:17,FINSEQ_3:55; then
     rng F1 = rng(F1|i) \/ rng <*F1.(i+1)*> by FINSEQ_1:31; then
     rng F1 = rng(F|i) \/ {F1.(i+1)} by A9,FINSEQ_1:38; then
     rng F1 = rng(F|i) \/ {F.(i+1)} by FINSEQ_3:112; then
     union rng F1 = union rng(F|i) \/ union {F.(i+1)} by ZFMISC_1:78; then
     Union F1 = union rng(F|i) \/ union {F.(i+1)} by CARD_3:def 4; then
     Union F1 = Union (F|i) \/ union {F.(i+1)} by CARD_3:def 4; then
A11: Union F1 = Union (F|i) \/ F.(i+1) by ZFMISC_1:25;

     i+1 in dom F by A8,NAT_1:12,FINSEQ_3:25; then
     F.(i+1) in rng F by FUNCT_1:3; then
     F.(i+1) in S;
     hence P[i+1] by A1,A5,A11,FINSUB_1:def 1;
    end;
   end;
   for i be Nat holds P[i] from NAT_1:sch 2(A3,A4);
   hence thesis;
end;
