reserve X for non empty TopSpace,
  A,B for Subset of X;
reserve Y1,Y2 for non empty SubSpace of X;

theorem Th8:
  for X being non trivial TopSpace, Y1 being proper non
empty SubSpace of X ex Y2 being proper strict non empty SubSpace of X st Y1,Y2
  constitute_a_decomposition
proof
  let X be non trivial TopSpace, Y1 be proper non empty SubSpace
  of X;
  reconsider A1 = the carrier of Y1 as Subset of X by TSEP_1:1;
  A1 is proper by TEX_2:8;
  then consider Y2 being strict non empty SubSpace of X such that
A1: A1` = the carrier of Y2 by TSEP_1:10;
A2: for P, Q be Subset of X st P = the carrier of Y1 & Q = the carrier of Y2
  holds P,Q constitute_a_decomposition by A1,TSEP_2:5;
  then Y1,Y2 constitute_a_decomposition;
  then reconsider Y2 as proper strict non empty SubSpace of X by Th6;
  take Y2;
  thus thesis by A2;
end;
