
theorem lemma103:
for X being set, S being with_empty_element semi-diff-closed cap-closed
  Subset-Family of X, A,B being set st A in S & B in S
holds B \ A in DisUnion S
proof
   let X be set, S be with_empty_element semi-diff-closed cap-closed
   Subset-Family of X, A,B be set;
   assume
A2: A in S & B in S;
   reconsider A1=A,B1=B as Subset of X by A2;
   ex F be disjoint_valued FinSequence of S st B\A = Union F by A2,DefSD; then
   B1 \ A1 in DisUnion S;
   hence thesis;
end;
