let X be non empty TopSpace; :: thesis: for X1, X0, X2 being non empty SubSpace of X st X1,X0 constitute_a_decomposition & X0,X2 constitute_a_decomposition holds
TopStruct(# the carrier of X1,the topology of X1 #) = TopStruct(# the carrier of X2,the topology of X2 #)
let X1, X0, X2 be non empty SubSpace of X; :: thesis: ( X1,X0 constitute_a_decomposition & X0,X2 constitute_a_decomposition implies TopStruct(# the carrier of X1,the topology of X1 #) = TopStruct(# the carrier of X2,the topology of X2 #) )
assume A1: for A1, A0 being Subset of X st A1 = the carrier of X1 & A0 = the carrier of X0 holds
A1,A0 constitute_a_decomposition ; :: according to TSEP_2:def 2 :: thesis: ( not X0,X2 constitute_a_decomposition or TopStruct(# the carrier of X1,the topology of X1 #) = TopStruct(# the carrier of X2,the topology of X2 #) )
assume A2: for A0, A2 being Subset of X st A0 = the carrier of X0 & A2 = the carrier of X2 holds
A0,A2 constitute_a_decomposition ; :: according to TSEP_2:def 2 :: thesis: TopStruct(# the carrier of X1,the topology of X1 #) = TopStruct(# the carrier of X2,the topology of X2 #)
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,A0 constitute_a_decomposition & A0,A2 constitute_a_decomposition ) by A1, A2;
hence TopStruct(# the carrier of X1,the topology of X1 #) = TopStruct(# the carrier of X2,the topology of X2 #) by Th9, TSEP_1:5; :: thesis: verum