let R be Ring; :: thesis: for V being RightMod of R
for u, v being Vector of V
for W being Submodule of V st v + W = u + W holds
ex v1 being Vector of V st
( v1 in W & v + v1 = u )
let V be RightMod of R; :: thesis: for u, v being Vector of V
for W being Submodule of V st v + W = u + W holds
ex v1 being Vector of V st
( v1 in W & v + v1 = u )
let u, v be Vector of V; :: thesis: for W being Submodule of V st v + W = u + W holds
ex v1 being Vector of V st
( v1 in W & v + v1 = u )
let W be Submodule of V; :: thesis: ( v + W = u + W implies ex v1 being Vector of V st
( v1 in W & v + v1 = u ) )
assume v + W = u + W ; :: thesis: ex v1 being Vector of V st
( v1 in W & v + v1 = u )
then v in u + W by Th44;
then consider u1 being Vector 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:def 4 ; :: thesis: verum