let X be non empty TopSpace; :: thesis: for X1, X2 being non empty SubSpace of X st X1,X2 constitute_a_decomposition holds
( X1 is open iff X2 is closed )
let X1, X2 be non empty SubSpace of X; :: thesis: ( X1,X2 constitute_a_decomposition implies ( X1 is open iff X2 is closed ) )
reconsider A1 = the carrier of X1, A2 = the carrier of X2 as Subset of X by TSEP_1:1;
assume A1: for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 constitute_a_decomposition ; :: according to TSEP_2:def 2 :: thesis: ( X1 is open iff X2 is closed )
thus ( X1 is open implies X2 is closed ) :: thesis: ( X2 is closed implies X1 is open )

proof

assume A2: for A1 being Subset of X st A1 = the carrier of X1 holds
A1 is open ; :: according to TSEP_1:def 1 :: thesis: X2 is closed

now

let A2 be Subset of X; :: thesis: ( A2 = the carrier of X2 implies A2 is closed )
assume A2 = the carrier of X2 ; :: thesis: A2 is closed
then A3: A1,A2 constitute_a_decomposition by A1;
A1 is open by A2;
hence A2 is closed by A3, Th12; :: thesis: verum

end;

hence X2 is closed by BORSUK_1:def 11; :: thesis: verum

end;

assume A4: for A2 being Subset of X st A2 = the carrier of X2 holds
A2 is closed ; :: according to BORSUK_1:def 11 :: thesis: X1 is open

now

let A1 be Subset of X; :: thesis: ( A1 = the carrier of X1 implies A1 is open )
assume A1 = the carrier of X1 ; :: thesis: A1 is open
then A5: A1,A2 constitute_a_decomposition by A1;
A2 is closed by A4;
hence A1 is open by A5, Th12; :: thesis: verum

end;

hence X1 is open by TSEP_1:def 1; :: thesis: verum