let V be RealLinearSpace; :: thesis: 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; :: thesis: 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; :: thesis: ( 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 #) ; :: thesis: 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; :: thesis: verum

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; :: thesis: 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; :: thesis: ( 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 #) ; :: thesis: 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; :: thesis: verum