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 C being Coset of W holds
( u in C iff C = u + W )
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 C being Coset of W holds
( u in C iff C = u + W )
let u be Element of V; :: thesis: for W being Subspace of V
for C being Coset of W holds
( u in C iff C = u + W )
let W be Subspace of V; :: thesis: for C being Coset of W holds
( u in C iff C = u + W )
let C be Coset of W; :: thesis: ( u in C iff C = u + W )
thus ( u in C implies C = u + W ) :: thesis: ( C = u + W implies u in C )

proof

assume A1: u in C ; :: thesis: C = u + W
ex v being Element of V st C = v + W by Def6;
hence C = u + W by A1, Th55; :: thesis: verum

end;

thus ( C = u + W implies u in C ) by Th44; :: thesis: verum