let X be non empty TopSpace; 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; ( 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
; TSEP_2:def 2 ( 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
; TSEP_2:def 2 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; verum