let X be non empty TopSpace; :: thesis: for X1, X2, Y1, Y2 being non empty SubSpace of X st X1 meets Y1 & X1,X2 constitute_a_decomposition & Y1,Y2 constitute_a_decomposition holds
X1 meet Y1,X2 union Y2 constitute_a_decomposition
let X1, X2, Y1, Y2 be non empty SubSpace of X; :: thesis: ( X1 meets Y1 & X1,X2 constitute_a_decomposition & Y1,Y2 constitute_a_decomposition implies X1 meet Y1,X2 union Y2 constitute_a_decomposition )
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 A1: X1 meets Y1 ; :: thesis: ( not X1,X2 constitute_a_decomposition or not Y1,Y2 constitute_a_decomposition or X1 meet Y1,X2 union Y2 constitute_a_decomposition )
assume 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: ( not Y1,Y2 constitute_a_decomposition or X1 meet Y1,X2 union Y2 constitute_a_decomposition )
then A2: A1,A2 constitute_a_decomposition ;
assume for B1, B2 being Subset of X st B1 = the carrier of Y1 & B2 = the carrier of Y2 holds
B1,B2 constitute_a_decomposition ; :: according to TSEP_2:def 2 :: thesis: X1 meet Y1,X2 union Y2 constitute_a_decomposition
then A3: B1,B2 constitute_a_decomposition ;
now :: thesis: for C, D being Subset of X st C = the carrier of (X1 meet Y1) & D = the carrier of (X2 union Y2) holds
C,D constitute_a_decomposition
let C, D be Subset of X; :: thesis: ( C = the carrier of (X1 meet Y1) & D = the carrier of (X2 union Y2) implies C,D constitute_a_decomposition )
assume ( C = the carrier of (X1 meet Y1) & D = the carrier of (X2 union Y2) ) ; :: thesis: C,D constitute_a_decomposition
then ( C = A1 /\ B1 & D = A2 \/ B2 ) by A1, TSEP_1:def 2, TSEP_1:def 4;
hence C,D constitute_a_decomposition by A2, A3, Th13; :: thesis: verum
end;
hence X1 meet Y1,X2 union Y2 constitute_a_decomposition ; :: thesis: verum