reserve V,X,Y for RealLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a for Real;
reserve V1,V2,V3 for Subset of V;
reserve x for object;
reserve W,W1,W2 for Subspace of V;
reserve w,w1,w2 for VECTOR of W;
reserve D for non empty set;
reserve d1 for Element of D;
reserve A for BinOp of D;
reserve M for Function of [:REAL,D:],D;
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;
  end;
  - u in W iff v + W = (v + (- u)) + W by Th52;
  hence thesis by A1,Th22;
end;
