let V be RealLinearSpace; for W being Subspace of V
for L being Linear_Compl of W holds
( W + L = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) & L + W = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) )
let W be Subspace of V; for L being Linear_Compl of W holds
( W + L = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) & L + W = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) )
let L be Linear_Compl of W; ( W + L = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) & L + W = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) )
V is_the_direct_sum_of W,L
by Th35;
hence
W + L = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
; L + W = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
hence
L + W = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
by Lm1; verum