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 u being Element of V

for W being Subspace of V

for B, C being Coset of W st u in B & u in C holds

B = C

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 u being Element of V

for W being Subspace of V

for B, C being Coset of W st u in B & u in C holds

B = C

let u be Element of V; :: thesis: for W being Subspace of V

for B, C being Coset of W st u in B & u in C holds

B = C

let W be Subspace of V; :: thesis: for B, C being Coset of W st u in B & u in C holds

B = C

let B, C be Coset of W; :: thesis: ( u in B & u in C implies B = C )

assume A1: ( u in B & u in C ) ; :: thesis: B = C

( ex v1 being Element of V st B = v1 + W & ex v2 being Element of V st C = v2 + W ) by Def6;

hence B = C by A1, Th56; :: thesis: verum

for u being Element of V

for W being Subspace of V

for B, C being Coset of W st u in B & u in C holds

B = C

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 u being Element of V

for W being Subspace of V

for B, C being Coset of W st u in B & u in C holds

B = C

let u be Element of V; :: thesis: for W being Subspace of V

for B, C being Coset of W st u in B & u in C holds

B = C

let W be Subspace of V; :: thesis: for B, C being Coset of W st u in B & u in C holds

B = C

let B, C be Coset of W; :: thesis: ( u in B & u in C implies B = C )

assume A1: ( u in B & u in C ) ; :: thesis: B = C

( ex v1 being Element of V st B = v1 + W & ex v2 being Element of V st C = v2 + W ) by Def6;

hence B = C by A1, Th56; :: thesis: verum