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 )

assume TopStruct(# the carrier of X, the topology of X #) = X1 union X2 ; :: thesis: X1,X2 constitute_a_decomposition

then for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds

A1,A2 constitute_a_decomposition by A3, TSEP_1:def 2;

hence X1,X2 constitute_a_decomposition ; :: thesis: verum

( 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 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
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 ;

hence X1 misses X2 ; :: thesis: TopStruct(# the carrier of X, the topology of X #) = X1 union X2

A1 \/ A2 = the carrier of X by A1;

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;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 ;

hence X1 misses X2 ; :: thesis: TopStruct(# the carrier of X, the topology of X #) = X1 union X2

A1 \/ A2 = the carrier of X by A1;

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

assume TopStruct(# the carrier of X, the topology of X #) = X1 union X2 ; :: thesis: X1,X2 constitute_a_decomposition

then for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds

A1,A2 constitute_a_decomposition by A3, TSEP_1:def 2;

hence X1,X2 constitute_a_decomposition ; :: thesis: verum