reserve X for non empty 1-sorted;
reserve A, A1, A2, B1, B2 for Subset of X;
reserve X for non empty TopSpace;
reserve A, A1, A2, B1, B2 for Subset of X;
reserve X for non empty 1-sorted;
reserve A, A1, A2, B1, B2 for Subset of X;
reserve X for non empty TopSpace,
  A1, A2 for Subset of X;
reserve X0 for non empty SubSpace of X,
  B1, B2 for Subset of X0;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;

theorem
  X1,X0 constitute_a_decomposition & X0,X2 constitute_a_decomposition
  implies the TopStruct of X1 = the TopStruct of X2
proof
  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;
  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;
  then A0,A2 constitute_a_decomposition;
  hence thesis by A1,Th8,TSEP_1:5;
end;
