let V be RealLinearSpace; :: thesis: for W being Subspace of V holds (0. V) + W = the carrier of W
let W be Subspace 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: { ((0. V) + u) where u is VECTOR of V : u in W } c= the carrier of W

proof

let x be set ; :: 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
A2: x = (0. V) + u and
A3: u in W ;
x = u by A2, RLVECT_1:10;
hence x in the carrier of W by A3, STRUCT_0:def 5; :: thesis: verum

end;

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

proof

let x be set ; :: 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 A4: x in W by STRUCT_0:def 5;
then x in V by Th17;
then reconsider y = x as Element of V by STRUCT_0:def 5;
(0. V) + y = x by RLVECT_1:10;
hence x in { ((0. V) + u) where u is VECTOR of V : u in W } by A4; :: thesis: verum

end;

hence (0. V) + W = the carrier of W by A1, XBOOLE_0:def 10; :: thesis: verum