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
  v - u in v + W iff u in W
proof
A1: v - u = (- u) + v;
A2: - u in W implies - (- u) in W by Th22;
  u in W implies - u in W by Th22;
  hence thesis by A1,A2,Th57,RLVECT_1:17;
end;
