reserve X for non empty TopSpace,
  A,B for Subset of X;
reserve Y1,Y2 for non empty SubSpace of X;
reserve X1, X2 for non empty SubSpace of X;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X for non discrete non empty TopSpace;

theorem
  for X1 being boundary non empty SubSpace of X ex X2 being dense proper
  strict non empty SubSpace of X st X1,X2 constitute_a_decomposition
proof
  let X1 be boundary non empty SubSpace of X;
  consider X2 being proper strict non empty SubSpace of X such that
A1: X1,X2 constitute_a_decomposition by Th8;
  reconsider X2 as dense proper strict non empty SubSpace of X by A1,Th31;
  take X2;
  thus thesis by A1;
end;
