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 Th61:
  v + W = u + W implies ex v1 st v1 in W & v + v1 = u
proof
  assume v + W = u + W;
  then v in u + W by Th44;
  then consider u1 such that
A1: v = u + u1 and
A2: u1 in W;
  take v1 = u - v;
  0.V = (u + u1) - v by A1,VECTSP_1:19
    .= u1 + (u - v) by RLVECT_1:def 3;
  then v1 = - u1 by VECTSP_1:16;
  hence v1 in W by A2,Th22;
  thus v + v1 = (u + v) - v by RLVECT_1:def 3
    .= u + (v - v) by RLVECT_1:def 3
    .= u + 0.V by VECTSP_1:19
    .= u by RLVECT_1:def 4;
end;
