reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X for TopSpace;
reserve A1, A2 for Subset of X;
reserve A1,A2 for Subset of X;
reserve X for TopSpace,
  A1, A2 for Subset of X;
reserve X for non empty TopSpace,
  A1, A2 for Subset of X;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X1, X2 for non empty SubSpace of X;

theorem Th81:
  for X1, X2 being open SubSpace of X holds X1,X2 are_weakly_separated
proof
  let X1, X2 be open SubSpace of X;
  let A1, A2 be Subset of X;
  reconsider B1 = A1, B2 = A2 as Subset of X;
  assume A1 = the carrier of X1 & A2 = the carrier of X2;
  then B1 is open & B2 is open by Th16;
  hence A1,A2 are_weakly_separated by Th49;
end;
