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 Th62:
  v + W = u + W implies ex v1 st v1 in W & v - v1 = u
proof
  assume v + W = u + W;
  then u in v + W by Th44;
  then consider u1 such that
A1: u = v + u1 and
A2: u1 in W;
  take v1 = v - u;
  0.V = (v + u1) - u by A1,VECTSP_1:19
    .= u1 + (v - u) by RLVECT_1:def 3;
  then v1 = - u1 by VECTSP_1:16;
  hence v1 in W by A2,Th22;
  thus v - v1 = (v - v) + u by RLVECT_1:29
    .= 0.V + u by VECTSP_1:19
    .= u by RLVECT_1:def 4;
end;
