let V be RealLinearSpace; :: thesis: for u, v being VECTOR of V
for W being Subspace 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 Subspace of V st v + W = u + W holds
ex v1 being VECTOR of V st
( v1 in W & v + v1 = u )
let W be Subspace 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 Th43;
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, RLVECT_1:15
.= u1 + (u - v) by RLVECT_1:def 3 ;
then v1 = - u1 by RLVECT_1:def 10;
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 RLVECT_1:15
.= u ; :: thesis: verum