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 being Subspace of M
for W2 being strict Subspace of M st the carrier of W1 c= the carrier of W2 holds
W1 + W2 = 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 being Subspace of M
for W2 being strict Subspace of M st the carrier of W1 c= the carrier of W2 holds
W1 + W2 = W2
let W1 be Subspace of M; :: thesis: for W2 being strict Subspace of M st the carrier of W1 c= the carrier of W2 holds
W1 + W2 = W2
let W2 be strict Subspace of M; :: thesis: ( the carrier of W1 c= the carrier of W2 implies W1 + W2 = W2 )
assume A1: the carrier of W1 c= the carrier of W2 ; :: thesis: W1 + W2 = W2
A2: the carrier of (W1 + W2) c= the carrier of W2
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in the carrier of (W1 + W2) or x in the carrier of W2 )
assume x in the carrier of (W1 + W2) ; :: thesis: x in the carrier of W2
then x in { (v + u) where u, v is Element of M : ( v in W1 & u in W2 ) } by Def1;
then consider u, v being Element of M such that
A3: x = v + u and
A4: v in W1 and
A5: u in W2 ;
W1 is Subspace of W2 by A1, VECTSP_4:27;
then v in W2 by A4, VECTSP_4:8;
then v + u in W2 by A5, VECTSP_4:20;
hence x in the carrier of W2 by A3, STRUCT_0:def 5; :: thesis: verum
end;
W1 + W2 = W2 + W1 by Lm1;
then the carrier of W2 c= the carrier of (W1 + W2) by Lm2;
then the carrier of (W1 + W2) = the carrier of W2 by A2;
hence W1 + W2 = W2 by VECTSP_4:29; :: thesis: verum