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

W1 + (W2 /\ W3) = (W1 + W2) /\ W3

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

assume A1: W1 is strict Subspace of W3 ; :: thesis: W1 + (W2 /\ W3) = (W1 + W2) /\ W3

thus (W1 + W2) /\ W3 = W3 /\ (W1 + W2) by Th14

.= (W1 /\ W3) + (W3 /\ W2) by A1, Lm9, RLSUB_1:30

.= W1 + (W3 /\ W2) by A1, Th17

.= W1 + (W2 /\ W3) by Th14 ; :: thesis: verum

W1 + (W2 /\ W3) = (W1 + W2) /\ W3

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

assume A1: W1 is strict Subspace of W3 ; :: thesis: W1 + (W2 /\ W3) = (W1 + W2) /\ W3

thus (W1 + W2) /\ W3 = W3 /\ (W1 + W2) by Th14

.= (W1 /\ W3) + (W3 /\ W2) by A1, Lm9, RLSUB_1:30

.= W1 + (W3 /\ W2) by A1, Th17

.= W1 + (W2 /\ W3) by Th14 ; :: thesis: verum