let V be RealUnitarySpace; :: thesis: for W being Subspace of V
for L being Linear_Compl of W holds
( W + L = UNITSTR(# the carrier of V,the ZeroF of V,the U7 of V,the Mult of V,the scalar of V #) & L + W = UNITSTR(# the carrier of V,the ZeroF of V,the U7 of V,the Mult of V,the scalar of V #) )
let W be Subspace of V; :: thesis: for L being Linear_Compl of W holds
( W + L = UNITSTR(# the carrier of V,the ZeroF of V,the U7 of V,the Mult of V,the scalar of V #) & L + W = UNITSTR(# the carrier of V,the ZeroF of V,the U7 of V,the Mult of V,the scalar of V #) )
let L be Linear_Compl of W; :: thesis: ( W + L = UNITSTR(# the carrier of V,the ZeroF of V,the U7 of V,the Mult of V,the scalar of V #) & L + W = UNITSTR(# the carrier of V,the ZeroF of V,the U7 of V,the Mult of V,the scalar of V #) )
V is_the_direct_sum_of W,L by Th35;
hence W + L = UNITSTR(# the carrier of V,the ZeroF of V,the U7 of V,the Mult of V,the scalar of V #) by Def4; :: thesis: L + W = UNITSTR(# the carrier of V,the ZeroF of V,the U7 of V,the Mult of V,the scalar of V #)
hence L + W = UNITSTR(# the carrier of V,the ZeroF of V,the U7 of V,the Mult of V,the scalar of V #) by Lm1; :: thesis: verum