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
  v - u in v + W iff u in W
proof
A1: v - u = (- u) + v;
A2: - u in W implies - (- u) in W by Th22;
  u in W implies - u in W by Th22;
  hence thesis by A1,A2,Th60,RLVECT_1:17;
end;
