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 /\ W2) + W2) = the carrier of 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 /\ W2) + W2) = the carrier of W2
let W1, W2 be Subspace of M; :: thesis: the carrier of ((W1 /\ W2) + W2) = the carrier of W2
thus the carrier of ((W1 /\ W2) + W2) c= the carrier of W2 :: according to XBOOLE_0:def 10 :: thesis: the carrier of W2 c= the carrier of ((W1 /\ W2) + W2)
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in the carrier of ((W1 /\ W2) + W2) or x in the carrier of W2 )
assume x in the carrier of ((W1 /\ W2) + W2) ; :: thesis: x in the carrier of W2
then x in { (u + v) where v, u is Element of M : ( u in W1 /\ W2 & v in W2 ) } by Def1;
then consider v, u being Element of M such that
A1: x = u + v and
A2: u in W1 /\ W2 and
A3: v in W2 ;
u in W2 by A2, Th3;
then u + v in W2 by A3, VECTSP_4:20;
hence x in the carrier of W2 by A1, STRUCT_0:def 5; :: thesis: verum
end;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in the carrier of W2 or x in the carrier of ((W1 /\ W2) + W2) )
the carrier of W2 c= the carrier of (W2 + (W1 /\ W2)) by Lm2;
then A4: the carrier of W2 c= the carrier of ((W1 /\ W2) + W2) by Lm1;
assume x in the carrier of W2 ; :: thesis: x in the carrier of ((W1 /\ W2) + W2)
hence x in the carrier of ((W1 /\ W2) + W2) by A4; :: thesis: verum