theorem Th3:
  for A, B being set st A in F & B in F holds A \/ B in F
proof
  let A, B be set;
  assume
A1: A in F & B in F;
  then reconsider A1 = A, B1 = B as Subset of X;
  A1` in F & B1` in F by A1,Def1;
  then A1` /\ B1` in F by FINSUB_1:def 2;
  then (A1 \/ B1)` in F by XBOOLE_1:53;
  then (A1 \/ B1)`` in F by Def1;
  hence thesis;
end;
