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
  the carrier of V is Coset of (Omega).V
proof
  set v = the Vector of V;
  the carrier of V c= the carrier of V;
  then reconsider A = the carrier of V as Subset of V;
  A = v + (Omega).V by Th47;
  hence thesis by Def6;
end;
