let X, Y be set ; :: thesis: ( ( X is constituted-DTrees & Y is constituted-DTrees ) iff X \/ Y is constituted-DTrees )
thus ( X is constituted-DTrees & Y is constituted-DTrees implies X \/ Y is constituted-DTrees ) :: thesis: ( X \/ Y is constituted-DTrees implies ( X is constituted-DTrees & Y is constituted-DTrees ) )

proof

assume that
A1: for x being object st x in X holds
x is DecoratedTree and
A2: for x being object st x in Y holds
x is DecoratedTree ; :: according to TREES_3:def 5 :: thesis: X \/ Y is constituted-DTrees
let x be object ; :: according to TREES_3:def 5 :: thesis: ( x in X \/ Y implies x is DecoratedTree )
assume x in X \/ Y ; :: thesis: x is DecoratedTree
then ( x in X or x in Y ) by XBOOLE_0:def 3;
hence x is DecoratedTree by A1, A2; :: thesis: verum

end;

assume A3: for x being object st x in X \/ Y holds
x is DecoratedTree ; :: according to TREES_3:def 5 :: thesis: ( X is constituted-DTrees & Y is constituted-DTrees )
thus X is constituted-DTrees :: thesis: Y is constituted-DTrees

proof

let x be object ; :: according to TREES_3:def 5 :: thesis: ( x in X implies x is DecoratedTree )
assume x in X ; :: thesis: x is DecoratedTree
then x in X \/ Y by XBOOLE_0:def 3;
hence x is DecoratedTree by A3; :: thesis: verum

end;

let x be object ; :: according to TREES_3:def 5 :: thesis: ( x in Y implies x is DecoratedTree )
assume x in Y ; :: thesis: x is DecoratedTree
then x in X \/ Y by XBOOLE_0:def 3;
hence x is DecoratedTree by A3; :: thesis: verum