let V be RealLinearSpace; :: thesis: for W being Subspace of V

for L being Linear_Compl of W holds

( W /\ L = (0). V & L /\ W = (0). V )

let W be Subspace of V; :: thesis: for L being Linear_Compl of W holds

( W /\ L = (0). V & L /\ W = (0). V )

let L be Linear_Compl of W; :: thesis: ( W /\ L = (0). V & L /\ W = (0). V )

V is_the_direct_sum_of W,L by Th35;

hence W /\ L = (0). V ; :: thesis: L /\ W = (0). V

hence L /\ W = (0). V by Th14; :: thesis: verum

for L being Linear_Compl of W holds

( W /\ L = (0). V & L /\ W = (0). V )

let W be Subspace of V; :: thesis: for L being Linear_Compl of W holds

( W /\ L = (0). V & L /\ W = (0). V )

let L be Linear_Compl of W; :: thesis: ( W /\ L = (0). V & L /\ W = (0). V )

V is_the_direct_sum_of W,L by Th35;

hence W /\ L = (0). V ; :: thesis: L /\ W = (0). V

hence L /\ W = (0). V by Th14; :: thesis: verum