
theorem Th9:
  for V being Abelian add-associative right_zeroed
  right_complementable non empty addLoopStr, M,N being Subset of V, v being
  Element of V holds M = N + {v} iff M - {v} = N
proof
  let V be Abelian add-associative right_zeroed right_complementable non
  empty addLoopStr;
  let M,N be Subset of V;
  let v be Element of V;
A1: M - {v} = N implies M = N + {v}
  proof
    assume
A2: M - {v} = N;
    for x being object st x in N + {v} holds x in M
    proof
      let x be object;
      assume
A3:   x in N + {v};
      then reconsider x as Element of V;
      x in {u1 + v1 where u1,v1 is Element of V : u1 in N & v1 in {v}} by A3,
RUSUB_4:def 9;
      then consider u1,v1 being Element of V such that
A4:   x = u1 + v1 and
A5:   u1 in N and
A6:   v1 in {v};
A7:   x - v1 = u1 + (v1 - v1) by A4,RLVECT_1:def 3
        .= u1 + 0.V by RLVECT_1:15
        .= u1;
      v1 = v by A6,TARSKI:def 1;
      then consider u2,v2 being Element of V such that
A8:   x - v = u2 - v2 and
A9:   u2 in M and
A10:  v2 in {v} by A2,A5,A7;
      v2 = v by A10,TARSKI:def 1;
      then x - v + v = u2 - (v - v) by A8,RLVECT_1:29
        .= u2 - 0.V by RLVECT_1:15
        .= u2;
      then u2 = x - (v - v) by RLVECT_1:29
        .= x - 0.V by RLVECT_1:15;
      hence thesis by A9;
    end;
    then
A11: N + {v} c= M;
    for x being object st x in M holds x in N + {v}
    proof
      let x be object;
      assume
A12:  x in M;
      then reconsider x as Element of V;
A13:  v in {v} by TARSKI:def 1;
      then
      x - v in {u2 - v2 where u2,v2 is Element of V : u2 in M & v2 in {v}
      } by A12;
      then consider u2 being Element of V such that
A14:  x - v = u2 and
A15:  u2 in N by A2;
      u2 + v = x - (v - v) by A14,RLVECT_1:29
        .= x - 0.V by RLVECT_1:15
        .= x;
      then
      x in {u1 + v1 where u1,v1 is Element of V : u1 in N & v1 in {v}} by A13
,A15;
      hence thesis by RUSUB_4:def 9;
    end;
    then M c= N + {v};
    hence thesis by A11;
  end;
  M = N + {v} implies M - {v} = N
  proof
    assume
A16: M = N + {v};
    for x being object st x in M - {v} holds x in N
    proof
      let x be object;
      assume
A17:  x in M - {v};
      then reconsider x as Element of V;
      consider u1,v1 being Element of V such that
A18:  x = u1 - v1 and
A19:  u1 in M and
A20:  v1 in {v} by A17;
A21:  x + v1 = u1 - (v1 - v1) by A18,RLVECT_1:29
        .= u1 - 0.V by RLVECT_1:15
        .= u1;
      v1 = v by A20,TARSKI:def 1;
      then
      x + v in {u2 + v2 where u2,v2 is Element of V : u2 in N & v2 in {v}
      } by A16,A19,A21,RUSUB_4:def 9;
      then consider u2,v2 being Element of V such that
A22:  x + v = u2 + v2 & u2 in N and
A23:  v2 in {v};
      v2 = v by A23,TARSKI:def 1;
      hence thesis by A22,RLVECT_1:8;
    end;
    then
A24: M - {v} c= N;
    for x being object st x in N holds x in M - {v}
    proof
      let x be object;
      assume
A25:  x in N;
      then reconsider x as Element of V;
A26:  v in {v} by TARSKI:def 1;
      then
      x + v in {u2 + v2 where u2,v2 is Element of V : u2 in N & v2 in {v}
      } by A25;
      then x + v in M by A16,RUSUB_4:def 9;
      then consider u2 being Element of V such that
A27:  x + v = u2 and
A28:  u2 in M;
      u2 - v = x + (v - v) by A27,RLVECT_1:def 3
        .= x + 0.V by RLVECT_1:15
        .= x;
      hence thesis by A26,A28;
    end;
    then N c= M - {v};
    hence thesis by A24;
  end;
  hence thesis by A1;
end;
