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 Th51:
  u in W iff v + W = (v + u) + W
proof
  thus u in W implies v + W = (v + u) + W
  proof
    assume
A1: u in W;
    thus v + W c= (v + u) + W
    proof
      let x be object;
      assume x in v + W;
      then consider v1 such that
A2:   x = v + v1 and
A3:   v1 in W;
A4:   (v + u) + (v1 - u) = v + (u + (v1 - u)) by RLVECT_1:def 3
        .= v + ((v1 + u) - u) by RLVECT_1:def 3
        .= v + (v1 + (u - u)) by RLVECT_1:def 3
        .= v + (v1 + 0.V) by VECTSP_1:19
        .= x by A2,RLVECT_1:def 4;
      v1 - u in W by A1,A3,Th23;
      hence thesis by A4;
    end;
    let x be object;
    assume x in (v + u) + W;
    then consider v2 such that
A5: x = (v + u) + v2 and
A6: v2 in W;
A7: x = v + (u + v2) by A5,RLVECT_1:def 3;
    u + v2 in W by A1,A6,Th20;
    hence thesis by A7;
  end;
  assume
A8: v + W = (v + u) + W;
  0.V in W & v + 0.V = v by Th17,RLVECT_1:def 4;
  then v in (v + u) + W by A8;
  then consider u1 such that
A9: v = (v + u) + u1 and
A10: u1 in W;
  v = v + (u + u1) by A9,RLVECT_1:def 3;
  then u + u1 = 0.V by RLVECT_1:9;
  then u = - u1 by VECTSP_1:16;
  hence thesis by A10,Th22;
end;
