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 ;

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

hence
X1 meet Y1,X2 union Y2 constitute_a_decomposition
; :: thesis: verumC,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;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