let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2 being Subspace of M holds the carrier of W1 c= the carrier of (W1 + W2)
let M be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for W1, W2 being Subspace of M holds the carrier of W1 c= the carrier of (W1 + W2)
let W1, W2 be Subspace of M; :: thesis: the carrier of W1 c= the carrier of (W1 + W2)
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in the carrier of W1 or x in the carrier of (W1 + W2) )
set A = { (v + u) where u, v is Element of M : ( v in W1 & u in W2 ) } ;
assume x in the carrier of W1 ; :: thesis: x in the carrier of (W1 + W2)
then reconsider v = x as Element of W1 ;
reconsider v = v as Element of M by VECTSP_4:10;
A1: v = v + (0. M) by RLVECT_1:4;
( v in W1 & 0. M in W2 ) by STRUCT_0:def 5, VECTSP_4:17;
then x in { (v + u) where u, v is Element of M : ( v in W1 & u in W2 ) } by A1;
hence x in the carrier of (W1 + W2) by Def1; :: thesis: verum