let X be TopSpace; :: thesis: ( X is almost_discrete iff for A being Subset of X st A is closed holds
Int A = A )
thus ( X is almost_discrete implies for A being Subset of X st A is closed holds
Int A = A ) :: thesis: ( ( for A being Subset of X st A is closed holds
Int A = A ) implies X is almost_discrete )

proof

assume A1: X is almost_discrete ; :: thesis: for A being Subset of X st A is closed holds
Int A = A
let A be Subset of X; :: thesis: ( A is closed implies Int A = A )
assume A is closed ; :: thesis: Int A = A
then A is open by A1, Th24;
hence Int A = A by TOPS_1:23; :: thesis: verum

end;

assume A2: for A being Subset of X st A is closed holds
Int A = A ; :: thesis: X is almost_discrete

now

let A be Subset of X; :: thesis: ( A is closed implies A is open )
assume A is closed ; :: thesis: A is open
then Int A = A by A2;
hence A is open ; :: thesis: verum

end;

hence X is almost_discrete by Th24; :: thesis: verum