let A be Subset of T; :: thesis: ( A is 2nd_class implies ( not A is 1st_class & not A is 3rd_class ) )
assume A is 2nd_class ; :: thesis: ( not A is 1st_class & not A is 3rd_class )
then A3: Cl (Int A) c< Int (Cl A) by Def7;
then A4: ( Cl (Int A) c= Int (Cl A) & Cl (Int A) <> Int (Cl A) ) by XBOOLE_0:def 8;
A5: not Int (Cl A) c= Cl (Int A) by A3, XBOOLE_0:def 8;
Cl (Int A), Int (Cl A) are_c=-comparable by A4, XBOOLE_0:def 9;
hence ( not A is 1st_class & not A is 3rd_class ) by A5, Def6, Def8; :: thesis: verum