let F be Field; :: thesis: for V being VectSp of F
for W being Subspace of V
for L being Linear_Compl of W holds
( W + L = ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) & L + W = ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) )
let V be VectSp of F; :: thesis: for W being Subspace of V
for L being Linear_Compl of W holds
( W + L = ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) & L + W = ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) )
let W be Subspace of V; :: thesis: for L being Linear_Compl of W holds
( W + L = ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) & L + W = ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) )
let L be Linear_Compl of W; :: thesis: ( W + L = ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) & L + W = ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) )
V is_the_direct_sum_of W,L by Th38;
hence W + L = ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) ; :: thesis: L + W = ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #)
hence L + W = ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) by Lm1; :: thesis: verum