reserve X for set;
reserve S for Subset-Family of X;

theorem thmCup:
  for S be cap-finite-partition-closed diff-finite-partition-closed
  Subset-Family of X holds
  {union x where x is finite Subset of S:x is mutually-disjoint}
  is cup-closed
  proof
    let S be cap-finite-partition-closed diff-finite-partition-closed
    Subset-Family of X;
    set Y={union x where x is finite Subset of S:x is mutually-disjoint};
    let a,b be set;
    assume
A:  a in Y & b in Y;
    then consider a0 be finite Subset of S such that
B:  a = union a0 & a0 is mutually-disjoint;
    consider b0 be finite Subset of S such that
C:  b = union b0 & b0 is mutually-disjoint by A;
    consider SM be FinSequence such that
F1: rng SM=a0\/b0 by FINSEQ_1:52;
    consider P be finite Subset of S such that
VU: P is a_partition of Union SM and
    for Y be Element of (rng SM) holds
    Y = union {s where s is Element of S: s in P & s c= Y} by F1,Thm87;
    Union SM=union a0 \/ union b0 by F1,ZFMISC_1:78;
    then
VB: union P = a\/b by B,C,VU,EQREL_1:def 4;
    for x,y be set st x in P & y in P & x<>y holds x misses y
    by VU,EQREL_1:def 4;
    then P is mutually-disjoint by TAXONOM2:def 5;
    hence thesis by VB;
  end;
