let X be non empty TopSpace; :: thesis: for X1, X0, X2 being non empty SubSpace of X st X1 is SubSpace of X0 & ( X0 misses X2 or X2 misses X0 ) holds
( X0 meet (X1 union X2) = TopStruct(# the carrier of X1,the topology of X1 #) & X0 meet (X2 union X1) = TopStruct(# the carrier of X1,the topology of X1 #) )
let X1, X0, X2 be non empty SubSpace of X; :: thesis: ( X1 is SubSpace of X0 & ( X0 misses X2 or X2 misses X0 ) implies ( X0 meet (X1 union X2) = TopStruct(# the carrier of X1,the topology of X1 #) & X0 meet (X2 union X1) = TopStruct(# the carrier of X1,the topology of X1 #) ) )
reconsider A0 = the carrier of X0, A1 = the carrier of X1, A2 = the carrier of X2 as Subset of X by TSEP_1:1;
assume A1: X1 is SubSpace of X0 ; :: thesis: ( ( not X0 misses X2 & not X2 misses X0 ) or ( X0 meet (X1 union X2) = TopStruct(# the carrier of X1,the topology of X1 #) & X0 meet (X2 union X1) = TopStruct(# the carrier of X1,the topology of X1 #) ) )
then ( X0 meets X1 & X1 is SubSpace of X1 union X2 ) by Th22, TSEP_1:22;
then A2: X0 meets X1 union X2 by Th23;
A3: A1 c= A0 by A1, TSEP_1:4;
assume ( X0 misses X2 or X2 misses X0 ) ; :: thesis: ( X0 meet (X1 union X2) = TopStruct(# the carrier of X1,the topology of X1 #) & X0 meet (X2 union X1) = TopStruct(# the carrier of X1,the topology of X1 #) )
then A4: A0 misses A2 by TSEP_1:def 3;
thus X0 meet (X1 union X2) = TopStruct(# the carrier of X1,the topology of X1 #) :: thesis: X0 meet (X2 union X1) = TopStruct(# the carrier of X1,the topology of X1 #)

proof

the carrier of (X0 meet (X1 union X2)) = A0 /\ the carrier of (X1 union X2) by A2, TSEP_1:def 5
.= A0 /\ (A1 \/ A2) by TSEP_1:def 2
.= (A0 /\ A1) \/ (A0 /\ A2) by XBOOLE_1:23
.= (A0 /\ A1) \/ {} by A4, XBOOLE_0:def 7
.= A1 by A3, XBOOLE_1:28 ;
hence X0 meet (X1 union X2) = TopStruct(# the carrier of X1,the topology of X1 #) by TSEP_1:5; :: thesis: verum

end;

hence X0 meet (X2 union X1) = TopStruct(# the carrier of X1,the topology of X1 #) ; :: thesis: verum