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 Th57:
  u + v in v + W iff u in W
proof
  thus u + v in v + W implies u in W
  proof
    assume u + v in v + W;
    then ex v1 st u + v = v + v1 & v1 in W;
    hence thesis by RLVECT_1:8;
  end;
  assume u in W;
  hence thesis;
end;
