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

theorem Thm86:
  for S be cap-finite-partition-closed Subset-Family of X
  holds
  {union x where x is finite Subset of S:x is mutually-disjoint}
  is cap-closed
  proof
    let S be cap-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 such that
H3: a in Y and
H4: b in Y;
    a /\ b in Y
    proof
      consider xa be finite Subset of S such that
V1:   a=union xa and
V2:   xa is mutually-disjoint by H3;
      consider xb be finite Subset of S such that
V3:   b=union xb and
V4:   xb is mutually-disjoint by H4;
      consider p be finite Subset of S such that
K2:   p is a_partition of union xa /\ union xb by V2,V4,V;
K3:   union p = a /\ b by V1,V3,K2,EQREL_1:def 4;
      for x,y be set st x in p & y in p & x<>y holds x misses y
      by K2,EQREL_1:def 4;
      then p is mutually-disjoint by TAXONOM2:def 5;
      hence thesis by K3;
    end;
    hence thesis;
  end;
