let R be Ring; :: thesis: for V being LeftMod of R
for W1, W2 being Submodule of V holds the carrier of ((W1 /\ W2) + W2) = the carrier of W2
let V be LeftMod of R; :: thesis: for W1, W2 being Submodule of V holds the carrier of ((W1 /\ W2) + W2) = the carrier of W2
let W1, W2 be Submodule of V; :: 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 Vector of V : ( u in W1 /\ W2 & v in W2 ) } by VECTSP_5:def 1;
then consider v, u being Vector of V such that
A1: x = u + v and
A2: u in W1 /\ W2 and
A3: v in W2 ;
u in W2 by A2, Th94;
then u + v in W2 by A3, Th36;
hence x in the carrier of W2 by A1; :: 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 ((W1 /\ W2) + W2) by Lm6;
hence ( not x in the carrier of W2 or x in the carrier of ((W1 /\ W2) + W2) ) ; :: thesis: verum