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

for C1 being Coset of W1

for C2 being Coset of W2 st C1 = C2 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

for C1 being Coset of W1

for C2 being Coset of W2 st C1 = C2 holds

W1 = W2

let W1, W2 be strict Subspace of V; :: thesis: for C1 being Coset of W1

for C2 being Coset of W2 st C1 = C2 holds

W1 = W2

let C1 be Coset of W1; :: thesis: for C2 being Coset of W2 st C1 = C2 holds

W1 = W2

let C2 be Coset of W2; :: thesis: ( C1 = C2 implies W1 = W2 )

( ex v1 being Element of V st C1 = v1 + W1 & ex v2 being Element of V st C2 = v2 + W2 ) by Def6;

hence ( C1 = C2 implies W1 = W2 ) by Th67; :: thesis: verum

for W1, W2 being strict Subspace of V

for C1 being Coset of W1

for C2 being Coset of W2 st C1 = C2 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

for C1 being Coset of W1

for C2 being Coset of W2 st C1 = C2 holds

W1 = W2

let W1, W2 be strict Subspace of V; :: thesis: for C1 being Coset of W1

for C2 being Coset of W2 st C1 = C2 holds

W1 = W2

let C1 be Coset of W1; :: thesis: for C2 being Coset of W2 st C1 = C2 holds

W1 = W2

let C2 be Coset of W2; :: thesis: ( C1 = C2 implies W1 = W2 )

( ex v1 being Element of V st C1 = v1 + W1 & ex v2 being Element of V st C2 = v2 + W2 ) by Def6;

hence ( C1 = C2 implies W1 = W2 ) by Th67; :: thesis: verum