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 Th64:
  for W1,W2 being strict Submodule of V holds v + W1 = u + W2 implies W1 = W2
proof
  let W1,W2 be strict Submodule of V;
  assume
A1: v + W1 = u + W2;
  set V2 = the carrier of W2;
  set V1 = the carrier of W1;
  assume
A2: W1 <> W2;
A3: now
    set x = the Element of V1 \ V2;
    assume V1 \ V2 <> {};
    then x in V1 by XBOOLE_0:def 5;
    then
A4: x in W1;
    then x in V by Th9;
    then reconsider x as Element of V;
    set z = v + x;
    z in u + W2 by A1,A4;
    then consider u1 such that
A5: z = u + u1 and
A6: u1 in W2;
    x = 0.V + x by RLVECT_1:def 4
      .= (v + (- v)) + x by VECTSP_1:19
      .= - v + (u + u1) by A5,RLVECT_1:def 3;
    then
A7: (v + (- v + (u + u1))) + W1 = v + W1 by A4,Th51;
    v + (- v + (u + u1)) = (v + (- v)) + (u + u1) by RLVECT_1:def 3
      .= 0.V + (u + u1) by VECTSP_1:19
      .= u + u1 by RLVECT_1:def 4;
    then (u + u1) + W2 = (u + u1) + W1 by A1,A6,A7,Th51;
    hence thesis by A2,Th63;
  end;
A8: now
    set x = the Element of V2 \ V1;
    assume V2 \ V1 <> {};
    then x in V2 by XBOOLE_0:def 5;
    then
A9: x in W2;
    then x in V by Th9;
    then reconsider x as Element of V;
    set z = u + x;
    z in v + W1 by A1,A9;
    then consider u1 such that
A10: z = v + u1 and
A11: u1 in W1;
    x = 0.V + x by RLVECT_1:def 4
      .= (u + (- u)) + x by VECTSP_1:19
      .= - u + (v + u1) by A10,RLVECT_1:def 3;
    then
A12: (u + (- u + (v + u1))) + W2 = u + W2 by A9,Th51;
    u + (- u + (v + u1)) = (u + (- u)) + (v + u1) by RLVECT_1:def 3
      .= 0.V + (v + u1) by VECTSP_1:19
      .= v + u1 by RLVECT_1:def 4;
    then (v + u1) + W1 = (v + u1) + W2 by A1,A11,A12,Th51;
    hence thesis by A2,Th63;
  end;
  V1 <> V2 by A2,Th29;
  then not V1 c= V2 or not V2 c= V1;
  hence thesis by A3,A8,XBOOLE_1:37;
end;
