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

proof

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

end;

assume A3: for x being object st x in X \/ Y holds
x is Tree ; :: according to TREES_3:def 3 :: thesis: ( X is constituted-Trees & Y is constituted-Trees )
thus X is constituted-Trees :: thesis: Y is constituted-Trees

proof

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

end;

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