let GF be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; :: thesis: for M being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr of 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 VectSp-like Abelian add-associative right_zeroed VectSpStr of 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)) )
assume A1: W1 is Subspace of W2 ; :: thesis: 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;
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 set ; :: 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 u1, u2 is Element of M : ( u1 in W1 & u2 in W2 ) } by Def1;
then consider u1, u2 being Element of M such that
A2: x = u1 + u2 and
A3: u1 in W1 and
A4: u2 in W2 ;
A5: u1 in the carrier of W1 by A3, STRUCT_0:def 5;
A6: u2 in the carrier of W2 by A4, STRUCT_0:def 5;
the carrier of W1 c= the carrier of W2 by A1, VECTSP_4:def 2;
then ( u1 in the carrier of W2 & V2 is linearly-closed ) by A5, VECTSP_4:41;
then u1 + u2 in V2 by A6, VECTSP_4:def 1;
then A7: u1 + u2 in W2 by STRUCT_0:def 5;
A8: 0. M in W1 /\ W3 by VECTSP_4:25;
(u1 + u2) + (0. M) = u1 + u2 by RLVECT_1:10;
then x in { (u + v) where u, v is Element of M : ( u in W2 & v in W1 /\ W3 ) } by A2, A7, A8;
hence x in the carrier of (W2 + (W1 /\ W3)) by Def1; :: thesis: verum