reserve i, j, m, n, k for Nat,
  x, y for set,
  K for Field,
  a,a1 for Element of K;

theorem Th1:
  for V be VectSp of K for W1,W2,W12 be Subspace of V for U1,U2 be
Subspace of W12 st U1 = W1 & U2 = W2 holds W1 /\ W2 = U1 /\ U2 & W1 + W2 = U1 +
  U2
proof
  let V be VectSp of K;
  let W1,W2,W12 be Subspace of V;
  let U1,U2 be Subspace of W12 such that
A1: U1 = W1 and
A2: U2 = W2;
  reconsider U12=U1/\U2 as Subspace of V by VECTSP_4:26;
A3: the carrier of U12 c= the carrier of (W1/\W2)
  proof
    let x be object;
    assume x in the carrier of U12;
    then x in U1/\U2;
    then x in U1 & x in U2 by VECTSP_5:3;
    then x in (W1/\W2) by A1,A2,VECTSP_5:3;
    hence thesis;
  end;
  the carrier of (W1/\W2) c= the carrier of U12
  proof
    let x be object;
    assume x in the carrier of (W1/\W2);
    then x in W1/\W2;
    then x in W1 & x in W2 by VECTSP_5:3;
    then x in U12 by A1,A2,VECTSP_5:3;
    hence thesis;
  end;
  then the carrier of (W1/\W2)=the carrier of U12 by A3,XBOOLE_0:def 10;
  hence W1/\W2=U1/\U2 by VECTSP_4:29;
  reconsider U12=U1+U2 as Subspace of V by VECTSP_4:26;
A4: the carrier of (W1+W2) c= the carrier of U12
  proof
    let x be object;
    assume x in the carrier of (W1+W2);
    then x in W1+W2;
    then consider v1,v2 be Vector of V such that
A5: v1 in W1 and
A6: v2 in W2 and
A7: v1+v2=x by VECTSP_5:1;
    U2 is Subspace of U12 by VECTSP_5:7;
    then
A8: v2 in U12 by A2,A6,VECTSP_4:8;
    U1 is Subspace of U12 by VECTSP_5:7;
    then v1 in U12 by A1,A5,VECTSP_4:8;
    then reconsider w1=v1,w2=v2 as Vector of U12 by A8;
    v1+v2=w1+w2 by VECTSP_4:13;
    hence thesis by A7;
  end;
  the carrier of U12 c= the carrier of (W1+W2)
  proof
    let x be object;
    assume x in the carrier of U12;
    then x in U1+U2;
    then consider v1,v2 be Vector of W12 such that
A9: v1 in U1 & v2 in U2 & v1+v2=x by VECTSP_5:1;
    reconsider w1=v1,w2=v2 as Vector of V by VECTSP_4:10;
    v1+v2=w1+w2 by VECTSP_4:13;
    then x in W1+W2 by A1,A2,A9,VECTSP_5:1;
    hence thesis;
  end;
  then the carrier of (W1+W2)=the carrier of U12 by A4,XBOOLE_0:def 10;
  hence thesis by VECTSP_4:29;
end;
