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 Th3:
  X is constituted-Trees & Y is constituted-Trees iff X \/
  Y is constituted-Trees
proof
  thus X is constituted-Trees & Y is constituted-Trees implies
  X \/ Y is constituted-Trees
  proof
    assume that
A1: for x st x in X holds x is Tree and
A2: for x st x in Y holds x is Tree;
    let x;
    assume x in X \/ Y;
    then x in X or x in Y by XBOOLE_0:def 3;
    hence thesis by A1,A2;
  end;
  assume
A3: for x st x in X \/ Y holds x is Tree;
  thus X is constituted-Trees
  proof
    let x;
    assume x in X;
    then x in X \/ Y by XBOOLE_0:def 3;
    hence thesis by A3;
  end;
  let x;
  assume x in Y;
  then x in X \/ Y by XBOOLE_0:def 3;
  hence thesis by A3;
end;
