let X be non empty TopSpace; :: thesis: for X1, X2, Y1, Y2 being non empty SubSpace of X st X1,Y1 constitute_a_decomposition & X2,Y2 constitute_a_decomposition holds
( Y1 union Y2 = TopStruct(# the carrier of X,the topology of X #) iff X1 misses X2 )
let X1, X2, Y1, Y2 be non empty SubSpace of X; :: thesis: ( X1,Y1 constitute_a_decomposition & X2,Y2 constitute_a_decomposition implies ( Y1 union Y2 = TopStruct(# the carrier of X,the topology of X #) iff X1 misses X2 ) )
reconsider A1 = the carrier of X1, A2 = the carrier of X2 as Subset of X by TSEP_1:1;
reconsider B1 = the carrier of Y1, B2 = the carrier of Y2 as Subset of X by TSEP_1:1;
assume ( X1,Y1 constitute_a_decomposition & X2,Y2 constitute_a_decomposition ) ; :: thesis: ( Y1 union Y2 = TopStruct(# the carrier of X,the topology of X #) iff X1 misses X2 )
then A1: ( A1,B1 constitute_a_decomposition & A2,B2 constitute_a_decomposition ) by Def2;
then A2: ( B1 = A1 ` & A1 = B1 ` & B2 = A2 ` & A2 = B2 ` ) by Th4;
A3: ( B1 = A1 ` & A1 = B1 ` & B2 = A2 ` & A2 = B2 ` ) by A1, Th4;
thus ( Y1 union Y2 = TopStruct(# the carrier of X,the topology of X #) implies X1 misses X2 ) :: thesis: ( X1 misses X2 implies Y1 union Y2 = TopStruct(# the carrier of X,the topology of X #) )

proof

assume Y1 union Y2 = TopStruct(# the carrier of X,the topology of X #) ; :: thesis: X1 misses X2
then B1 \/ B2 = the carrier of X by TSEP_1:def 2;
then (B1 \/ B2) ` = {} X by XBOOLE_1:37;
then A1 /\ A2 = {} by A3, XBOOLE_1:53;
then A1 misses A2 by XBOOLE_0:def 7;
hence X1 misses X2 by TSEP_1:def 3; :: thesis: verum

end;

assume X1 misses X2 ; :: thesis: Y1 union Y2 = TopStruct(# the carrier of X,the topology of X #)
then A1 misses A2 by TSEP_1:def 3;
then A1 /\ A2 = {} X by XBOOLE_0:def 7;
then (B1 \/ B2) ` = {} X by A2, XBOOLE_1:53;
then B1 \/ B2 = ({} X) ` ;
then ( the carrier of (Y1 union Y2) = the carrier of X & X is SubSpace of X ) by TSEP_1:2, TSEP_1:def 2;
hence Y1 union Y2 = TopStruct(# the carrier of X,the topology of X #) by TSEP_1:5; :: thesis: verum