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
  (ex C st v1 in C & v2 in C) iff v1 - v2 in W
proof
  thus (ex C st v1 in C & v2 in C) implies v1 - v2 in W
  proof
    given C such that
A1: v1 in C & v2 in C;
    ex v st C = v + W by Def6;
    hence thesis by A1,Th60;
  end;
  assume v1 - v2 in W;
  then consider v such that
A2: v1 in v + W & v2 in v + W by Th60;
  reconsider C = v + W as Coset of W by Def6;
  take C;
  thus thesis by A2;
end;
