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, v1, v2 being Element of V

for W being Subspace of V st u in v1 + W & u in v2 + W holds

v1 + W = v2 + 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, v1, v2 being Element of V

for W being Subspace of V st u in v1 + W & u in v2 + W holds

v1 + W = v2 + W

let u, v1, v2 be Element of V; :: thesis: for W being Subspace of V st u in v1 + W & u in v2 + W holds

v1 + W = v2 + W

let W be Subspace of V; :: thesis: ( u in v1 + W & u in v2 + W implies v1 + W = v2 + W )

assume that

A1: u in v1 + W and

A2: u in v2 + W ; :: thesis: v1 + W = v2 + W

thus v1 + W = u + W by A1, Th55

.= v2 + W by A2, Th55 ; :: thesis: verum

for u, v1, v2 being Element of V

for W being Subspace of V st u in v1 + W & u in v2 + W holds

v1 + W = v2 + 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, v1, v2 being Element of V

for W being Subspace of V st u in v1 + W & u in v2 + W holds

v1 + W = v2 + W

let u, v1, v2 be Element of V; :: thesis: for W being Subspace of V st u in v1 + W & u in v2 + W holds

v1 + W = v2 + W

let W be Subspace of V; :: thesis: ( u in v1 + W & u in v2 + W implies v1 + W = v2 + W )

assume that

A1: u in v1 + W and

A2: u in v2 + W ; :: thesis: v1 + W = v2 + W

thus v1 + W = u + W by A1, Th55

.= v2 + W by A2, Th55 ; :: thesis: verum