let F, G be ext-real-membered set ; :: thesis: -- (F \ G) = (-- F) \ (-- G)
let i be ext-real number ; :: according to MEMBERED:def 14 :: thesis: ( ( not i in -- (F \ G) or i in (-- F) \ (-- G) ) & ( not i in (-- F) \ (-- G) or i in -- (F \ G) ) )

hereby :: thesis: ( not i in (-- F) \ (-- G) or i in -- (F \ G) )

assume i in -- (F \ G) ; :: thesis: i in (-- F) \ (-- G)
then - i in F \ G by Th8;
then ( - i in F & not - i in G ) by XBOOLE_0:def 5;
then ( i in -- F & not i in -- G ) by Th8;
hence i in (-- F) \ (-- G) by XBOOLE_0:def 5; :: thesis: verum

end;

assume i in (-- F) \ (-- G) ; :: thesis: i in -- (F \ G)
then ( i in -- F & not i in -- G ) by XBOOLE_0:def 5;
then ( - i in F & not - i in G ) by Th8;
then - i in F \ G by XBOOLE_0:def 5;
hence i in -- (F \ G) by Th8; :: thesis: verum