let X be non empty TopSpace; :: thesis: ( X is extremally_disconnected iff for A being Subset of X st A is condensed holds
( A is closed_condensed & A is open_condensed ) )
thus ( X is extremally_disconnected implies for A being Subset of X st A is condensed holds
( A is closed_condensed & A is open_condensed ) ) by Th32, TOPS_1:66, TOPS_1:67; :: thesis: ( ( for A being Subset of X st A is condensed holds
( A is closed_condensed & A is open_condensed ) ) implies X is extremally_disconnected )
assume A1: for A being Subset of X st A is condensed holds
( A is closed_condensed & A is open_condensed ) ; :: thesis: X is extremally_disconnected

now :: thesis: for A being Subset of X st A is condensed holds
( A is closed & A is open )

let A be Subset of X; :: thesis: ( A is condensed implies ( A is closed & A is open ) )
assume A2: A is condensed ; :: thesis: ( A is closed & A is open )
then A3: A is open_condensed by A1;
A is closed_condensed by A1, A2;
hence ( A is closed & A is open ) by A3, TOPS_1:66, TOPS_1:67; :: thesis: verum

end;

hence X is extremally_disconnected by Th32; :: thesis: verum