let R be Ring; for V being RightMod of R
for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 holds
the carrier of (W2 + (W1 /\ W3)) = the carrier of ((W1 + W2) /\ (W2 + W3))
let V be RightMod of R; for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 holds
the carrier of (W2 + (W1 /\ W3)) = the carrier of ((W1 + W2) /\ (W2 + W3))
let W1, W2, W3 be Submodule of V; ( W1 is Submodule 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 V by RMOD_2:def 2;
A1:
V2 is linearly-closed
by RMOD_2:33;
assume
W1 is Submodule 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 RMOD_2:def 2;
thus
the carrier of (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (W2 + W3))
by Lm12; 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 Vector of V : ( u1 in W1 & u2 in W2 ) }
by Def1;
then consider u2, u1 being Vector of V 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;
then
u1 + u2 in V2
by A2, A1;
then A5:
u1 + u2 in W2
;
( 0. V in W1 /\ W3 & (u1 + u2) + (0. V) = u1 + u2 )
by RLVECT_1:def 4, RMOD_2:17;
then
x in { (u + v) where v, u is Vector of V : ( u in W2 & v in W1 /\ W3 ) }
by A3, A5;
hence
x in the carrier of (W2 + (W1 /\ W3))
by Def1; verum