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, v being Element of V
for W being Subspace of V
for C being Coset of W st u in C & v in C holds
ex v1 being Element of V st
( v1 in W & u + v1 = v )

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, v being Element of V
for W being Subspace of V
for C being Coset of W st u in C & v in C holds
ex v1 being Element of V st
( v1 in W & u + v1 = v )

let u, v be Element of V; :: thesis: for W being Subspace of V
for C being Coset of W st u in C & v in C holds
ex v1 being Element of V st
( v1 in W & u + v1 = v )

let W be Subspace of V; :: thesis: for C being Coset of W st u in C & v in C holds
ex v1 being Element of V st
( v1 in W & u + v1 = v )

let C be Coset of W; :: thesis: ( u in C & v in C implies ex v1 being Element of V st
( v1 in W & u + v1 = v ) )

assume ( u in C & v in C ) ; :: thesis: ex v1 being Element of V st
( v1 in W & u + v1 = v )

then ( C = u + W & C = v + W ) by Th77;
hence ex v1 being Element of V st
( v1 in W & u + v1 = v ) by Th64; :: thesis: verum