let V be RealLinearSpace; :: thesis: for W1, W2, W3 being Subspace of V st W1 is Subspace of W3 holds

W1 /\ W2 is Subspace of W3

let W1, W2, W3 be Subspace of V; :: thesis: ( W1 is Subspace of W3 implies W1 /\ W2 is Subspace of W3 )

A1: W1 /\ W2 is Subspace of W1 by RLSUB_2:16;

assume W1 is Subspace of W3 ; :: thesis: W1 /\ W2 is Subspace of W3

hence W1 /\ W2 is Subspace of W3 by A1, RLSUB_1:27; :: thesis: verum

W1 /\ W2 is Subspace of W3

let W1, W2, W3 be Subspace of V; :: thesis: ( W1 is Subspace of W3 implies W1 /\ W2 is Subspace of W3 )

A1: W1 /\ W2 is Subspace of W1 by RLSUB_2:16;

assume W1 is Subspace of W3 ; :: thesis: W1 /\ W2 is Subspace of W3

hence W1 /\ W2 is Subspace of W3 by A1, RLSUB_1:27; :: thesis: verum