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 Th53:
  v in u + W iff u + W = v + W
proof
  thus v in u + W implies u + W = v + W
  proof
    assume v in u + W;
    then consider z being Vector of V such that
A1: v = u + z and
A2: z in W;
    thus u + W c= v + W
    proof
      let x be object;
      assume x in u + W;
      then consider v1 such that
A3:   x = u + v1 and
A4:   v1 in W;
      v - z = u + (z - z) by A1,RLVECT_1:def 3
        .= u + 0.V by VECTSP_1:19
        .= u by RLVECT_1:def 4;
      then
A5:   x = v + (v1 + (- z)) by A3,RLVECT_1:def 3
        .= v + (v1 - z);
      v1 - z in W by A2,A4,Th23;
      hence thesis by A5;
    end;
    let x be object;
    assume x in v + W;
    then consider v2 such that
A6: x = v + v2 & v2 in W;
    z + v2 in W & x = u + (z + v2) by A1,A2,A6,Th20,RLVECT_1:def 3;
    hence thesis;
  end;
  thus thesis by Th44;
end;
