 reserve R for Ring;
 reserve x, y, y1 for set;
 reserve a, b for Element of R;
 reserve V for LeftMod of R;
 reserve v, w for Vector of V;
 reserve u,v,w for Vector of V;
 reserve F,G,H,I for FinSequence of V;
 reserve j,k,n for Nat;
 reserve f,f9,g for sequence of V;
 reserve R for Ring;
 reserve V, X, Y for LeftMod of R;
 reserve u, u1, u2, v, v1, v2 for Vector of V;
 reserve a for Element of R;
 reserve V1, V2, V3 for Subset of V;
 reserve x for set;
 reserve W, W1, W2 for Submodule 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 [:the carrier of R,D:],D;
reserve B,C for Coset of W;

theorem
  u in v + W iff ex v1 st v1 in W & u = v + v1
  proof
    thus u in v + W implies ex v1 st v1 in W & u = v + v1
    proof
      assume u in v + W;
      then ex v1 st u = v + v1 & v1 in W;
      hence thesis;
    end;
    given v1 such that
A1: v1 in W & u = v + v1;
    thus thesis by A1;
  end;
