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 strict Subspace of V st ( for v being Element of V holds

( v in W1 iff v in W2 ) ) holds

W1 = 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 strict Subspace of V st ( for v being Element of V holds

( v in W1 iff v in W2 ) ) holds

W1 = W2

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

( v in W1 iff v in W2 ) ) implies W1 = W2 )

assume A1: for v being Element of V holds

( v in W1 iff v in W2 ) ; :: thesis: W1 = W2

for x being object holds

( x in the carrier of W1 iff x in the carrier of W2 )

hence W1 = W2 by Th29; :: thesis: verum

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

( v in W1 iff v in W2 ) ) holds

W1 = 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 strict Subspace of V st ( for v being Element of V holds

( v in W1 iff v in W2 ) ) holds

W1 = W2

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

( v in W1 iff v in W2 ) ) implies W1 = W2 )

assume A1: for v being Element of V holds

( v in W1 iff v in W2 ) ; :: thesis: W1 = W2

for x being object holds

( x in the carrier of W1 iff x in the carrier of W2 )

proof

then
the carrier of W1 = the carrier of W2
by TARSKI:2;
let x be object ; :: thesis: ( x in the carrier of W1 iff x in the carrier of W2 )

thus ( x in the carrier of W1 implies x in the carrier of W2 ) :: thesis: ( x in the carrier of W2 implies x in the carrier of W1 )

the carrier of W2 c= the carrier of V by Def2;

then reconsider v = x as Element of V by A3;

v in W2 by A3;

then v in W1 by A1;

hence x in the carrier of W1 ; :: thesis: verum

end;thus ( x in the carrier of W1 implies x in the carrier of W2 ) :: thesis: ( x in the carrier of W2 implies x in the carrier of W1 )

proof

assume A3:
x in the carrier of W2
; :: thesis: x in the carrier of W1
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;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

the carrier of W2 c= the carrier of V by Def2;

then reconsider v = x as Element of V by A3;

v in W2 by A3;

then v in W1 by A1;

hence x in the carrier of W1 ; :: thesis: verum

hence W1 = W2 by Th29; :: thesis: verum