let X be non empty TopSpace; :: thesis: for X0, X1, 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 X0, X1, 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 #) )
reconsider A0 = the carrier of X0, A1 = the carrier of X1, A2 = the carrier of X2 as Subset of X by TSEP_1:1;
assume 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 #) )
then A1: A1,A0 constitute_a_decomposition ;
assume 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 #)
then A0,A2 constitute_a_decomposition ;
hence TopStruct(# the carrier of X1, the topology of X1 #) = TopStruct(# the carrier of X2, the topology of X2 #) by A1, Th8, TSEP_1:5; :: thesis: verum