let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for W being Subspace of V holds (0. V) + W = the carrier of W

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: 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 Element of V : u in W } ;

A1: the carrier of W c= { ((0. V) + u) where u is Element of V : u in W }

for W being Subspace of V holds (0. V) + W = the carrier of W

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: 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 Element of V : u in W } ;

A1: the carrier of W c= { ((0. V) + u) where u is Element of V : u in W }

proof

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

assume x in the carrier of W ; :: thesis: x in { ((0. V) + u) where u is Element 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_1:4;

hence x in { ((0. V) + u) where u is Element of V : u in W } by A2; :: thesis: verum

end;assume x in the carrier of W ; :: thesis: x in { ((0. V) + u) where u is Element 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_1:4;

hence x in { ((0. V) + u) where u is Element of V : u in W } by A2; :: thesis: verum

proof

hence
(0. V) + W = the carrier of W
by A1; :: thesis: verum
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { ((0. V) + u) where u is Element of V : u in W } or x in the carrier of W )

assume x in { ((0. V) + u) where u is Element of V : u in W } ; :: thesis: x in the carrier of W

then consider u being Element of V such that

A3: x = (0. V) + u and

A4: u in W ;

x = u by A3, RLVECT_1:4;

hence x in the carrier of W by A4; :: thesis: verum

end;assume x in { ((0. V) + u) where u is Element of V : u in W } ; :: thesis: x in the carrier of W

then consider u being Element of V such that

A3: x = (0. V) + u and

A4: u in W ;

x = u by A3, RLVECT_1:4;

hence x in the carrier of W by A4; :: thesis: verum