let V be RealLinearSpace; :: thesis: for v, u 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 v, u 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 A1: v + W = u + W ; :: thesis: ex v1 being VECTOR of V st
( v1 in W & v - v1 = u )
take v1 = v - u; :: thesis: ( v1 in W & v - v1 = u )
u in v + W by A1, Th59;
then consider u1 being VECTOR of V such that
A2: u = v + u1 and
A3: u1 in W ;
0. V = (v + u1) - u by A2, RLVECT_1:28
.= u1 + (v - u) by RLVECT_1:def 6 ;
then v1 = - u1 by RLVECT_1:def 11;
hence v1 in W by A3, Th30; :: thesis: v - v1 = u
thus v - v1 = (v - v) + u by RLVECT_1:43
.= (0. V) + u by RLVECT_1:28
.= u by RLVECT_1:10 ; :: thesis: verum