let R be Ring; :: thesis: for V being LeftMod of R
for W1, W2, W3 being Subspace of V holds W1 /\ W2 is Subspace of (W1 + W3) /\ W2
let V be LeftMod of R; :: thesis: for W1, W2, W3 being Subspace of V holds W1 /\ W2 is Subspace of (W1 + W3) /\ W2
let W1, W2, W3 be Subspace of V; :: thesis: W1 /\ W2 is Subspace of (W1 + W3) /\ W2
for v being Vector of V st v in W1 /\ W2 holds
v in (W1 + W3) /\ W2
proof
let v be Vector of V; :: thesis: ( v in W1 /\ W2 implies v in (W1 + W3) /\ W2 )
assume A1: v in W1 /\ W2 ; :: thesis: v in (W1 + W3) /\ W2
v in W1 by A1, VECTSP_5:3;
then A2: v in W1 + W3 by VECTSP_5:2;
v in W2 by A1, VECTSP_5:3;
hence v in (W1 + W3) /\ W2 by A2, VECTSP_5:3; :: thesis: verum
end;
hence W1 /\ W2 is Subspace of (W1 + W3) /\ W2 by VECTSP_4:28; :: thesis: verum