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 Th27:
  X1 meets X2 implies X1 meet X2 is SubSpace of X1 & X1 meet X2 is
  SubSpace of X2
proof
  assume X1 meets X2;
  then the carrier of X1 meet X2 = (the carrier of X1) /\ (the carrier of X2)
  by Def4;
  then
  the carrier of X1 meet X2 c= the carrier of X1 & the carrier of X1 meet
  X2 c= the carrier of X2 by XBOOLE_1:17;
  hence thesis by Th4;
end;
