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 st v + W = u + W holds

ex v1 being Element of V st

( v1 in W & v + v1 = u )

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 st v + W = u + W holds

ex v1 being Element of V st

( v1 in W & v + v1 = u )

let u, v be Element of V; :: thesis: for W being Subspace of V st v + W = u + W holds

ex v1 being Element of V st

( v1 in W & v + v1 = u )

let W be Subspace of V; :: thesis: ( v + W = u + W implies ex v1 being Element of V st

( v1 in W & v + v1 = u ) )

assume v + W = u + W ; :: thesis: ex v1 being Element of V st

( v1 in W & v + v1 = u )

then v in u + W by Th44;

then consider u1 being Element of V such that

A1: v = u + u1 and

A2: u1 in W ;

take v1 = u - v; :: thesis: ( v1 in W & v + v1 = u )

0. V = (u + u1) - v by A1, VECTSP_1:19

.= u1 + (u - v) by RLVECT_1:def 3 ;

then v1 = - u1 by VECTSP_1:16;

hence v1 in W by A2, Th22; :: thesis: v + v1 = u

thus v + v1 = (u + v) - v by RLVECT_1:def 3

.= u + (v - v) by RLVECT_1:def 3

.= u + (0. V) by VECTSP_1:19

.= u by RLVECT_1:4 ; :: thesis: verum

for u, v being Element of V

for W being Subspace of V st v + W = u + W holds

ex v1 being Element of V st

( v1 in W & v + v1 = u )

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 st v + W = u + W holds

ex v1 being Element of V st

( v1 in W & v + v1 = u )

let u, v be Element of V; :: thesis: for W being Subspace of V st v + W = u + W holds

ex v1 being Element of V st

( v1 in W & v + v1 = u )

let W be Subspace of V; :: thesis: ( v + W = u + W implies ex v1 being Element of V st

( v1 in W & v + v1 = u ) )

assume v + W = u + W ; :: thesis: ex v1 being Element of V st

( v1 in W & v + v1 = u )

then v in u + W by Th44;

then consider u1 being Element of V such that

A1: v = u + u1 and

A2: u1 in W ;

take v1 = u - v; :: thesis: ( v1 in W & v + v1 = u )

0. V = (u + u1) - v by A1, VECTSP_1:19

.= u1 + (u - v) by RLVECT_1:def 3 ;

then v1 = - u1 by VECTSP_1:16;

hence v1 in W by A2, Th22; :: thesis: v + v1 = u

thus v + v1 = (u + v) - v by RLVECT_1:def 3

.= u + (v - v) by RLVECT_1:def 3

.= u + (0. V) by VECTSP_1:19

.= u by RLVECT_1:4 ; :: thesis: verum