reserve x,y,z for object, X,Y for set,
  i,k,n for Nat,
  p,q,r,s for FinSequence,
  w for FinSequence of NAT,
  f for Function;

theorem
  X is constituted-FinTrees implies X /\ Y is constituted-FinTrees &
  Y /\ X is constituted-FinTrees & X \ Y is constituted-FinTrees
proof
  assume
A1: for x st x in X holds x is finite Tree;
  thus X /\ Y is constituted-FinTrees
  proof
    let x;
    assume x in X /\ Y;
    then x in X by XBOOLE_0:def 4;
    hence thesis by A1;
  end;
  hence Y /\ X is constituted-FinTrees;
  let x;
  assume x in X \ Y;
  hence thesis by A1;
end;
