let X be non empty TopSpace; :: thesis: for X1, X2 being SubSpace of X st X1,X2 constitute_a_decomposition holds
( X1 is dense iff X2 is boundary )
let X1, X2 be SubSpace of X; :: thesis: ( X1,X2 constitute_a_decomposition implies ( X1 is dense iff X2 is boundary ) )
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 dense iff X2 is boundary )
thus ( X1 is dense implies X2 is boundary ) :: thesis: ( X2 is boundary implies X1 is dense )
proof
assume A2: for A1 being Subset of X st A1 = the carrier of X1 holds
A1 is dense ; :: according to TEX_3:def 1 :: thesis: X2 is boundary
now :: thesis: for A2 being Subset of X st A2 = the carrier of X2 holds
A2 is boundary
let A2 be Subset of X; :: thesis: ( A2 = the carrier of X2 implies A2 is boundary )
assume A2 = the carrier of X2 ; :: thesis: A2 is boundary
then A3: A1,A2 constitute_a_decomposition by A1;
A1 is dense by A2;
hence A2 is boundary by A3, Th2; :: thesis: verum
end;
hence X2 is boundary ; :: thesis: verum
end;
assume A4: for A2 being Subset of X st A2 = the carrier of X2 holds
A2 is boundary ; :: according to TEX_3:def 3 :: thesis: X1 is dense
now :: thesis: for A1 being Subset of X st A1 = the carrier of X1 holds
A1 is dense
let A1 be Subset of X; :: thesis: ( A1 = the carrier of X1 implies A1 is dense )
assume A1 = the carrier of X1 ; :: thesis: A1 is dense
then A5: A1,A2 constitute_a_decomposition by A1;
A2 is boundary by A4;
hence A1 is dense by A5, Th2; :: thesis: verum
end;
hence X1 is dense ; :: thesis: verum