let T be TopSpace; :: thesis: for A being Subset of T holds
( A is subcondensed iff ( Cl A is regular_closed & Border A is empty ) )
let A be Subset of T; :: thesis: ( A is subcondensed iff ( Cl A is regular_closed & Border A is empty ) )
A1: Cl (Int A) c= Cl A by PRE_TOPC:19, TOPS_1:16;
thus ( A is subcondensed implies ( Cl A is regular_closed & Border A is empty ) ) :: thesis: ( Cl A is regular_closed & Border A is empty implies A is subcondensed )

proof

assume A2: A is subcondensed ; :: thesis: ( Cl A is regular_closed & Border A is empty )
then Cl (Int A) = Cl A ;
then Int (Cl A) c= Cl (Int A) by TOPS_1:16;
then (Int (Cl A)) \ (Cl (Int A)) is empty by XBOOLE_1:37;
hence ( Cl A is regular_closed & Border A is empty ) by A2, Th21; :: thesis: verum

end;

assume that
A3: Cl A is regular_closed and
A4: Border A is empty ; :: thesis: A is subcondensed
(Int (Cl A)) \ (Cl (Int A)) is empty by A4, Th21;
then Int (Cl A) c= Cl (Int A) by XBOOLE_1:37;
then A5: Cl (Int (Cl A)) c= Cl (Cl (Int A)) by PRE_TOPC:19;
Cl A = Cl (Int (Cl A)) by A3, TOPS_1:def 7;
then Cl (Int A) = Cl A by A5, A1, XBOOLE_0:def 10;
hence A is subcondensed ; :: thesis: verum