let Y be TopStruct ; :: thesis: ( ( for P, Q being Subset of Y st P is closed & Q is closed holds
( P /\ Q is closed & P \/ Q is closed ) ) implies for A, B being Subset of Y st A is closed & B is closed & A is discrete & B is discrete holds
A \/ B is discrete )
assume A1: for P, Q being Subset of Y st P is closed & Q is closed holds
( P /\ Q is closed & P \/ Q is closed ) ; :: thesis: for A, B being Subset of Y st A is closed & B is closed & A is discrete & B is discrete holds
A \/ B is discrete
let A, B be Subset of Y; :: thesis: ( A is closed & B is closed & A is discrete & B is discrete implies A \/ B is discrete )
assume A2: ( A is closed & B is closed ) ; :: thesis: ( not A is discrete or not B is discrete or A \/ B is discrete )
assume A3: ( A is discrete & B is discrete ) ; :: thesis: A \/ B is discrete

now

let D be Subset of Y; :: thesis: ( D c= A \/ B implies ex F being Subset of Y st
( F is closed & (A \/ B) /\ F = D ) )
assume D c= A \/ B ; :: thesis: ex F being Subset of Y st
( F is closed & (A \/ B) /\ F = D )
then A4: D = D /\ (A \/ B) by XBOOLE_1:28;
D /\ A c= A by XBOOLE_1:17;
then consider F1 being Subset of Y such that
A5: F1 is closed and
A6: A /\ F1 = D /\ A by A3, Def6;
D /\ B c= B by XBOOLE_1:17;
then consider F2 being Subset of Y such that
A7: F2 is closed and
A8: B /\ F2 = D /\ B by A3, Def6;

now

take F = (A /\ F1) \/ (B /\ F2); :: thesis: ( F is closed & (A \/ B) /\ F = D )
( A /\ F1 is closed & B /\ F2 is closed ) by A1, A2, A5, A7;
hence F is closed by A1; :: thesis: (A \/ B) /\ F = D
thus (A \/ B) /\ F = D by A4, A6, A8, XBOOLE_1:23; :: thesis: verum

end;

hence ex F being Subset of Y st
( F is closed & (A \/ B) /\ F = D ) ; :: thesis: verum

end;

hence A \/ B is discrete by Def6; :: thesis: verum