reserve x,y,y1,y2 for object;
reserve GF for add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr,
  V,X,Y for Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital non
  empty ModuleStr over GF;
reserve a for Element of GF;
reserve u,u1,u2,v,v1,v2 for Element of V;
reserve W,W1,W2 for Subspace of V;
reserve V1 for Subset of V;
reserve w,w1,w2 for Element of W;
reserve B,C for Coset of W;

theorem
  u in W iff v + W = (v - u) + W
proof
A1: - u in W implies u in W
  proof
    assume - u in W;
    then - (- u) in W by Th22;
    hence thesis by RLVECT_1:17;
  end;
  - u in W iff v + W = (v + (- u)) + W by Th53;
  hence thesis by A1,Th22;
end;
