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;

theorem
  X1 meets X2 implies X1 meet X2 is SubSpace of X1 union X2
proof
  assume X1 meets X2;
  then
A1: X1 meet X2 is SubSpace of X1 by Th27;
  X1 is SubSpace of X1 union X2 by Th22;
  hence thesis by A1,Th7;
end;
