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

theorem
  for X1,X2 being non empty SubSpace of X holds X1 is SubSpace of X2 iff
  X1 union X2 = the TopStruct of X2
proof
  let X1,X2 be non empty SubSpace of X;
  set A1 = the carrier of X1, A2 = the carrier of X2;
  thus X1 is SubSpace of X2 iff X1 union X2 = the TopStruct of X2
  proof
    thus X1 is SubSpace of X2 implies X1 union X2 = the TopStruct of X2
    proof
      assume X1 is SubSpace of X2;
      then
A1:   A1 \/ A2 = the carrier of the TopStruct of X2 by BORSUK_1:1,XBOOLE_1:12;
      the TopStruct of X2 is strict SubSpace of X by Lm3;
      hence thesis by A1,Def2;
    end;
    assume X1 union X2 = the TopStruct of X2;
    then A1 \/ A2 = A2 by Def2;
    then A1 c= A2 by XBOOLE_1:7;
    hence thesis by Th4;
  end;
end;
