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

for L being Linear_Compl of W holds W is Linear_Compl of L

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

let L be Linear_Compl of W; :: thesis: W is Linear_Compl of L

V is_the_direct_sum_of L,W by Def5;

then V is_the_direct_sum_of W,L by Lm16;

hence W is Linear_Compl of L by Def5; :: thesis: verum

for L being Linear_Compl of W holds W is Linear_Compl of L

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

let L be Linear_Compl of W; :: thesis: W is Linear_Compl of L

V is_the_direct_sum_of L,W by Def5;

then V is_the_direct_sum_of W,L by Lm16;

hence W is Linear_Compl of L by Def5; :: thesis: verum