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, W3 being Subspace of M st W1 is Subspace of W2 holds
the carrier of (W2 + (W1 /\ W3)) = the carrier of ((W1 + W2) /\ (W2 + W3))
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, W3 being Subspace of M st W1 is Subspace of W2 holds
the carrier of (W2 + (W1 /\ W3)) = the carrier of ((W1 + W2) /\ (W2 + W3))
let W1, W2, W3 be Subspace of M; :: thesis: ( W1 is Subspace of W2 implies the carrier of (W2 + (W1 /\ W3)) = the carrier of ((W1 + W2) /\ (W2 + W3)) )
reconsider V2 = the carrier of W2 as Subset of M by VECTSP_4:def 2;
A1: V2 is linearly-closed by VECTSP_4:33;
assume W1 is Subspace of W2 ; :: thesis: the carrier of (W2 + (W1 /\ W3)) = the carrier of ((W1 + W2) /\ (W2 + W3))
then A2: the carrier of W1 c= the carrier of W2 by VECTSP_4:def 2;
thus the carrier of (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (W2 + W3)) by Lm14; :: according to XBOOLE_0:def 10 :: thesis: the carrier of ((W1 + W2) /\ (W2 + W3)) c= the carrier of (W2 + (W1 /\ W3))
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in the carrier of ((W1 + W2) /\ (W2 + W3)) or x in the carrier of (W2 + (W1 /\ W3)) )
assume x in the carrier of ((W1 + W2) /\ (W2 + W3)) ; :: thesis: x in the carrier of (W2 + (W1 /\ W3))
then x in the carrier of (W1 + W2) /\ the carrier of (W2 + W3) by Def2;
then x in the carrier of (W1 + W2) by XBOOLE_0:def 4;
then x in { (u1 + u2) where u2, u1 is Element of M : ( u1 in W1 & u2 in W2 ) } by Def1;
then consider u2, u1 being Element of M such that
A3: x = u1 + u2 and
A4: ( u1 in W1 & u2 in W2 ) ;
( u1 in the carrier of W1 & u2 in the carrier of W2 ) by A4, STRUCT_0:def 5;
then u1 + u2 in V2 by A2, A1;
then A5: u1 + u2 in W2 by STRUCT_0:def 5;
( 0. M in W1 /\ W3 & (u1 + u2) + (0. M) = u1 + u2 ) by RLVECT_1:4, VECTSP_4:17;
then x in { (u + v) where v, u is Element of M : ( u in W2 & v in W1 /\ W3 ) } by A3, A5;
hence x in the carrier of (W2 + (W1 /\ W3)) by Def1; :: thesis: verum