let R be Ring; :: thesis: for V being RightMod of R
for W being Submodule of V holds (0. V) + W = the carrier of W
let V be RightMod of R; :: thesis: for W being Submodule of V holds (0. V) + W = the carrier of W
let W be Submodule of V; :: thesis: (0. V) + W = the carrier of W
set A = { ((0. V) + u) where u is Vector of V : u in W } ;
A1: the carrier of W c= { ((0. V) + u) where u is Vector of V : u in W }

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in the carrier of W or x in { ((0. V) + u) where u is Vector of V : u in W } )
assume x in the carrier of W ; :: thesis: x in { ((0. V) + u) where u is Vector of V : u in W }
then A2: x in W ;
then x in V by Th9;
then reconsider y = x as Element of V ;
(0. V) + y = x by RLVECT_1:def 4;
hence x in { ((0. V) + u) where u is Vector of V : u in W } by A2; :: thesis: verum

end;

{ ((0. V) + u) where u is Vector of V : u in W } c= the carrier of W

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { ((0. V) + u) where u is Vector of V : u in W } or x in the carrier of W )
assume x in { ((0. V) + u) where u is Vector of V : u in W } ; :: thesis: x in the carrier of W
then consider u being Vector of V such that
A3: x = (0. V) + u and
A4: u in W ;
x = u by A3, RLVECT_1:def 4;
hence x in the carrier of W by A4; :: thesis: verum

end;

hence (0. V) + W = the carrier of W by A1; :: thesis: verum