reserve x for set;
reserve a,b,c,d,e,r1,r2,r3,r4,r5,r6 for Real;
reserve V for RealLinearSpace;
reserve u,v,v1,v2,v3,w,w1,w2,w3 for VECTOR of V;
reserve W,W1,W2 for Subspace of V;

theorem
  u in v + W implies v + W = u + W
proof
  reconsider C = v + W as Coset of W by RLSUB_1:def 6;
  assume u in v + W;
  then C = u + W by RLSUB_1:78;
  hence thesis;
end;
