reserve x,y,y1,y2 for object;
reserve R for Ring;
reserve a for Scalar of R;
reserve V,X,Y for RightMod of R;
reserve u,u1,u2,v,v1,v2 for Vector of V;
reserve V1,V2,V3 for Subset of V;
reserve W,W1,W2 for Submodule of V;
reserve w,w1,w2 for Vector of W;
reserve B,C for Coset of W;

theorem
  for W1,W2 being strict Submodule of V for C1 being Coset of W1, C2
  being Coset of W2 holds C1 = C2 implies W1 = W2
proof
  let W1,W2 be strict Submodule of V;
  let C1 be Coset of W1, C2 be Coset of W2;
  ( ex v1 being Vector of V st C1 = v1 + W1)& ex v2 being Vector of V st
  C2 = v2 + W2 by Def6;
  hence thesis by Th64;
end;
