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 W1, W2 being Subspace of V st ( for v being Element of V st v in W1 holds

v in W2 ) holds

W1 is Subspace of W2

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 W1, W2 being Subspace of V st ( for v being Element of V st v in W1 holds

v in W2 ) holds

W1 is Subspace of W2

let W1, W2 be Subspace of V; :: thesis: ( ( for v being Element of V st v in W1 holds

v in W2 ) implies W1 is Subspace of W2 )

assume A1: for v being Element of V st v in W1 holds

v in W2 ; :: thesis: W1 is Subspace of W2

the carrier of W1 c= the carrier of W2

for W1, W2 being Subspace of V st ( for v being Element of V st v in W1 holds

v in W2 ) holds

W1 is Subspace of W2

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 W1, W2 being Subspace of V st ( for v being Element of V st v in W1 holds

v in W2 ) holds

W1 is Subspace of W2

let W1, W2 be Subspace of V; :: thesis: ( ( for v being Element of V st v in W1 holds

v in W2 ) implies W1 is Subspace of W2 )

assume A1: for v being Element of V st v in W1 holds

v in W2 ; :: thesis: W1 is Subspace of W2

the carrier of W1 c= the carrier of W2

proof

hence
W1 is Subspace of W2
by Th27; :: thesis: verum
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in the carrier of W1 or x in the carrier of W2 )

assume A2: x in the carrier of W1 ; :: thesis: x in the carrier of W2

the carrier of W1 c= the carrier of V by Def2;

then reconsider v = x as Element of V by A2;

v in W1 by A2;

then v in W2 by A1;

hence x in the carrier of W2 ; :: thesis: verum

end;assume A2: x in the carrier of W1 ; :: thesis: x in the carrier of W2

the carrier of W1 c= the carrier of V by Def2;

then reconsider v = x as Element of V by A2;

v in W1 by A2;

then v in W2 by A1;

hence x in the carrier of W2 ; :: thesis: verum