
theorem
for R being non degenerated Ring,
    S being Subset of R
holds ({S, {0.R}, -S} is a_partition of the carrier of R &
       S is add-closed mult-closed)
implies S \/ {0.R} is positive_cone
proof
let R be non degenerated Ring, S be Subset of R;
assume AS: ({S, {0.R}, -S} is a_partition of the carrier of R &
           S is add-closed mult-closed);
set T = S \/ {0.R};
J: now assume S = -S; then
    J1: {S, {0.R}} = {S, -S, {0.R}} by ENUMSET1:30
                  .= {S, {0.R}, -S} by ENUMSET1:57;
    J2: S in {S, {0.R}} & {0.R} in {S, {0.R}} by TARSKI:def 2;
    union {S, {0.R}} = the carrier of R by AS,J1,EQREL_1:def 4; then
    J3: S \/ {0.R} = the carrier of R by ZFMISC_1:75;
    not 1.R in {0.R} & not -1.R in {0.R} by TARSKI:def 1; then
    1.R in S & -1.R in S by J3,XBOOLE_0:def 3; then
    1.R + -1.R in S by AS; then
    0.R in S & 0.R in {0.R} by RLVECT_1:5,TARSKI:def 1; then
    0.R in S /\ {0.R}; then
    not S misses {0.R}; then
    S = {0.R} by AS,J1,J2,EQREL_1:def 4; then
    {S, {0.R}} = {{0.R}} by ENUMSET1:29; then
    union {{0.R}} = the carrier of R by AS,J1,EQREL_1:def 4;
    hence contradiction by ZFMISC_1:25;
    end;
I1: now assume AS1: S = {0.R};
    then -S = {0.R} by lempart0;
    then {S, {0.R}, -S} = {{0.R}} by AS1,ENUMSET1:36;
    then union {{0.R}} = the carrier of R by AS,EQREL_1:def 4;
    hence contradiction by ZFMISC_1:25;
    end;
    S in {S, {0.R}, -S} & {0.R} in {S, {0.R}, -S} by ENUMSET1:def 1;
    then S misses {0.R} by I1,AS,EQREL_1:def 4; then
K0: S \ {0.R} = S by XBOOLE_1:83;
I2: now assume AS2: -S = {0.R};
    then --S = -{0.R};
    then S = {0.R} by lempart0;
    then {S, {0.R}, -S} = {{0.R}} by AS2,ENUMSET1:36;
    then union {{0.R}} = the carrier of R by AS,EQREL_1:def 4;
    hence contradiction by ZFMISC_1:25;
    end;
    -S in {S, {0.R}, -S} & {0.R} in {S, {0.R}, -S} by ENUMSET1:def 1;
    then (-S) misses {0.R} by I2,AS,EQREL_1:def 4; then
K1: (-S) \ {0.R} = -S by XBOOLE_1:83;
H1: T^+ = S by K0,XBOOLE_1:40;
H2: T^- = ( (-S) \/ (-{0.R}) ) \ {0.R} by REALALG1:16
       .= ( (-S) \/ {0.R} ) \ {0.R} by lempart0
       .= -S by K1,XBOOLE_1:40;
   0.R in {0.R} by TARSKI:def 1; then
A: 0.R in T by XBOOLE_0:def 3;
T^+ /\ (T^-) = {}
   proof
   S in {S, {0.R}, -S} & -S in {S, {0.R}, -S} by ENUMSET1:def 1;
   then S misses (-S) by J,AS,EQREL_1:def 4;
   hence T^+ /\ (T^-) = {} by H2,K0,XBOOLE_1:40;
   end;
hence T is positive_cone by H1,H2,AS,A,bla;
end;
