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

W1 is Subspace of W2 /\ W3

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

assume A1: ( W1 is Subspace of W2 & W1 is Subspace of W3 ) ; :: thesis: W1 is Subspace of W2 /\ W3

W1 is Subspace of W2 /\ W3

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

assume A1: ( W1 is Subspace of W2 & W1 is Subspace of W3 ) ; :: thesis: W1 is Subspace of W2 /\ W3

now :: thesis: for v being VECTOR of V st v in W1 holds

v in W2 /\ W3

hence
W1 is Subspace of W2 /\ W3
by RLSUB_1:29; :: thesis: verumv in W2 /\ W3

let v be VECTOR of V; :: thesis: ( v in W1 implies v in W2 /\ W3 )

assume v in W1 ; :: thesis: v in W2 /\ W3

then ( v in W2 & v in W3 ) by A1, RLSUB_1:8;

hence v in W2 /\ W3 by RLSUB_2:3; :: thesis: verum

end;assume v in W1 ; :: thesis: v in W2 /\ W3

then ( v in W2 & v in W3 ) by A1, RLSUB_1:8;

hence v in W2 /\ W3 by RLSUB_2:3; :: thesis: verum