let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; 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; 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; ( 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
; 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; XBOOLE_0:def 10 the carrier of ((W1 + W2) /\ (W2 + W3)) c= the carrier of (W2 + (W1 /\ W3))
let x be object ; TARSKI:def 3 ( 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))
; 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; verum