let V be RealUnitarySpace; :: thesis: for W being Subspace of V
for u, v being VECTOR 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: for u, v being VECTOR 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: ( 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 u in v + W by Th37;
then consider u1 being VECTOR of V such that
A1: u = v + u1 and
A2: u1 in W ;
take v1 = v - u; :: thesis: ( v1 in W & v - v1 = u )
0. V = (v + u1) - u by A1, RLVECT_1:15
.= u1 + (v - u) by RLVECT_1:def 3 ;
then v1 = - u1 by RLVECT_1:def 10;
hence v1 in W by A2, Th16; :: thesis: v - v1 = u
thus v - v1 = (v - v) + u by RLVECT_1:29
.= (0. V) + u by RLVECT_1:15
.= u by RLVECT_1:4 ; :: thesis: verum