let X be non empty TopSpace; :: thesis: for X1, X2 being non empty SubSpace of X holds
( X1,X2 constitute_a_decomposition iff ( X1 misses X2 & TopStruct(# the carrier of X, the topology of X #) = X1 union X2 ) )
let X1, X2 be non empty SubSpace of X; :: thesis: ( X1,X2 constitute_a_decomposition iff ( X1 misses X2 & TopStruct(# the carrier of X, the topology of X #) = X1 union X2 ) )
reconsider A1 = the carrier of X1, A2 = the carrier of X2 as Subset of X by TSEP_1:1;
thus ( X1,X2 constitute_a_decomposition implies ( X1 misses X2 & TopStruct(# the carrier of X, the topology of X #) = X1 union X2 ) ) :: thesis: ( X1 misses X2 & TopStruct(# the carrier of X, the topology of X #) = X1 union X2 implies X1,X2 constitute_a_decomposition )

proof

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: ( X1 misses X2 & TopStruct(# the carrier of X, the topology of X #) = X1 union X2 )
then A1: A1,A2 constitute_a_decomposition ;
then A1 misses A2 by Def1;
hence X1 misses X2 by TSEP_1:def 3; :: thesis: TopStruct(# the carrier of X, the topology of X #) = X1 union X2
A1 \/ A2 = the carrier of X by A1, Def1;
then A2: the carrier of (X1 union X2) = the carrier of X by TSEP_1:def 2;
X is SubSpace of X by TSEP_1:2;
hence TopStruct(# the carrier of X, the topology of X #) = X1 union X2 by A2, TSEP_1:5; :: thesis: verum

end;

assume A3: X1 misses X2 ; :: thesis: ( not TopStruct(# the carrier of X, the topology of X #) = X1 union X2 or X1,X2 constitute_a_decomposition )
assume A4: TopStruct(# the carrier of X, the topology of X #) = X1 union X2 ; :: thesis: X1,X2 constitute_a_decomposition

now

let A1, A2 be Subset of X; :: thesis: ( A1 = the carrier of X1 & A2 = the carrier of X2 implies A1,A2 constitute_a_decomposition )
assume ( A1 = the carrier of X1 & A2 = the carrier of X2 ) ; :: thesis: A1,A2 constitute_a_decomposition
then ( A1 misses A2 & A1 \/ A2 = the carrier of X ) by A3, A4, TSEP_1:def 2, TSEP_1:def 3;
hence A1,A2 constitute_a_decomposition by Def1; :: thesis: verum

end;

hence X1,X2 constitute_a_decomposition by Def2; :: thesis: verum