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;

theorem
  for Y being non empty SubSpace of X st X1 meets X2 holds X1,Y
  are_separated implies X1 meet X2,Y are_separated
proof
  let Y be non empty SubSpace of X;
  assume X1 meets X2;
  then
A1: X1 meet X2 is SubSpace of X1 by Th27;
  Y is SubSpace of Y by Th2;
  hence thesis by A1,Th71;
end;
