let X be non empty TopSpace; :: thesis: for X0, X1, 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 X0, X1, 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;

A1: X1 is SubSpace of X1 union X2 by TSEP_1:22;

assume A2: 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 A3: A1 c= A0 by 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;

X0 meets X1 by A2, Th17;

then X0 meets X1 union X2 by A1, Th18;

then A5: the carrier of (X0 meet (X1 union X2)) = A0 /\ the carrier of (X1 union X2) by TSEP_1:def 4

.= A0 /\ (A1 \/ A2) by TSEP_1:def 2

.= (A0 /\ A1) \/ (A0 /\ A2) by XBOOLE_1:23

.= A1 by A3, XBOOLE_1:28, A4 ;

hence X0 meet (X1 union X2) = TopStruct(# the carrier of X1, the topology of X1 #) by TSEP_1:5; :: thesis: X0 meet (X2 union X1) = TopStruct(# the carrier of X1, the topology of X1 #)

thus X0 meet (X2 union X1) = TopStruct(# the carrier of X1, the topology of X1 #) by A5, TSEP_1:5; :: thesis: verum

( 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 X0, X1, 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;

A1: X1 is SubSpace of X1 union X2 by TSEP_1:22;

assume A2: 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 A3: A1 c= A0 by 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;

X0 meets X1 by A2, Th17;

then X0 meets X1 union X2 by A1, Th18;

then A5: the carrier of (X0 meet (X1 union X2)) = A0 /\ the carrier of (X1 union X2) by TSEP_1:def 4

.= A0 /\ (A1 \/ A2) by TSEP_1:def 2

.= (A0 /\ A1) \/ (A0 /\ A2) by XBOOLE_1:23

.= A1 by A3, XBOOLE_1:28, A4 ;

hence X0 meet (X1 union X2) = TopStruct(# the carrier of X1, the topology of X1 #) by TSEP_1:5; :: thesis: X0 meet (X2 union X1) = TopStruct(# the carrier of X1, the topology of X1 #)

thus X0 meet (X2 union X1) = TopStruct(# the carrier of X1, the topology of X1 #) by A5, TSEP_1:5; :: thesis: verum