let V be RealUnitarySpace; :: thesis: for W being Subspace of V
for v being VECTOR of V holds
( v in W iff v + W = the carrier of W )
let W be Subspace of V; :: thesis: for v being VECTOR of V holds
( v in W iff v + W = the carrier of W )
let v be VECTOR of V; :: thesis: ( v in W iff v + W = the carrier of W )
thus ( v in W implies v + W = the carrier of W ) :: thesis: ( v + W = the carrier of W implies v in W )

proof

assume A1: v in W ; :: thesis: v + W = the carrier of W
thus v + W c= the carrier of W :: according to XBOOLE_0:def 10 :: thesis: the carrier of W c= v + W

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in v + W or x in the carrier of W )
assume x in v + W ; :: thesis: x in the carrier of W
then consider u being VECTOR of V such that
A2: x = v + u and
A3: u in W ;
v + u in W by A1, A3, Th14;
hence x in the carrier of W by A2, STRUCT_0:def 5; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in the carrier of W or x in v + W )
assume x in the carrier of W ; :: thesis: x in v + W
then reconsider y = x, z = v as Element of W by A1, STRUCT_0:def 5;
reconsider y1 = y, z1 = z as VECTOR of V by Th3;
A4: y - z in W by STRUCT_0:def 5;
A5: z + (y - z) = (y + z) - z by RLVECT_1:def 6
.= y + (z - z) by RLVECT_1:def 6
.= y + (0. W) by RLVECT_1:28
.= x by RLVECT_1:10 ;
A6: y - z = y1 - z1 by Th10;
A7: y1 - z1 in W by A4, Th10;
z1 + (y1 - z1) = x by A5, A6, Th6;
hence x in v + W by A7; :: thesis: verum

end;

assume A8: v + W = the carrier of W ; :: thesis: v in W
assume A9: not v in W ; :: thesis: contradiction
( 0. V in W & v + (0. V) = v ) by Th11, RLVECT_1:10;
then v in { (v + u) where u is VECTOR of V : u in W } ;
hence contradiction by A8, A9, STRUCT_0:def 5; :: thesis: verum