
theorem
for R being non degenerated domRing,
    S being Subset of R
holds S is positive_cone
implies ({S^+, {0.R}, S^-} is a_partition of the carrier of R &
         S^+ is add-closed mult-closed)
proof
let R1 be non degenerated domRing, S be Subset of R1;
assume AS: S is positive_cone; then
reconsider R = R1 as ordered domRing by REALALG1:def 17;
reconsider P = S as Ordering of R by AS;
set M = {S^+, {0.R1}, S^-};
H: P \/ (-P) = the carrier of R & P /\ -P = {0.R} by REALALG1:def 15;
   now let o be object;
   assume o in the carrier of R; then
   per cases by H,XBOOLE_0:def 3;
   suppose o = 0.R;
     then o in {0.R} & {0.R} in M by TARSKI:def 1,ENUMSET1:def 1;
     hence o in union M by TARSKI:def 4;
     end;
   suppose o <> 0.R & o in P;
     then not o in {0.R} & o in P by TARSKI:def 1;
     then o in P^+ & P^+ in M by XBOOLE_0:def 5,ENUMSET1:def 1;
     hence o in union M by TARSKI:def 4;
     end;
   suppose o <> 0.R & o in -P;
     then not o in {0.R} & o in -P by TARSKI:def 1;
     then o in P^- & P^- in M by XBOOLE_0:def 5,ENUMSET1:def 1;
     hence o in union M by TARSKI:def 4;
     end;
   end; then
   the carrier of R c= union M; then
A: union M = the carrier of R;
   now let A be Subset of the carrier of R;
     assume B: A in M;
     thus A <> {}
       proof
       per cases by B,ENUMSET1:def 1;
       suppose A = {0.R};
         hence thesis;
         end;
       suppose A = P^+;
         hence thesis;
         end;
       suppose A = P^-;
         hence thesis;
         end;
       end;
     thus for B being Subset of the carrier of R st B in M
          holds A = B or A misses B
       proof
       let B be Subset of the carrier of R;
       assume C: B in M;
       assume D: A <> B;
       set x = the Element of A /\ B;
       per cases by B,ENUMSET1:def 1;
       suppose E: A = {0.R}; then
         per cases by C,D,ENUMSET1:def 1;
         suppose F: B = P^+;
           now assume A /\ B <> {};
             then x in {0.R} & x in B by E,XBOOLE_0:def 4;
             hence contradiction by F,XBOOLE_0:def 5;
             end;
           hence A misses B;
           end;
         suppose F: B = P^-;
           now assume A /\ B <> {};
             then x in {0.R} & x in B by E,XBOOLE_0:def 4;
             hence contradiction by F,XBOOLE_0:def 5;
             end;
           hence A misses B;
           end;
         end;
       suppose E: A = P^+; then
         per cases by C,D,ENUMSET1:def 1;
         suppose F: B = {0.R};
           now assume A /\ B <> {};
             then x in {0.R} & x in A by F,XBOOLE_0:def 4;
             hence contradiction by E,XBOOLE_0:def 5;
             end;
           hence A misses B;
           end;
         suppose B = P^-;
           hence A misses B by E;
           end;
         end;
       suppose E: A = P^-; then
         per cases by C,D,ENUMSET1:def 1;
         suppose F: B = {0.R};
           now assume A /\ B <> {};
             then x in {0.R} & x in A by F,XBOOLE_0:def 4;
             hence contradiction by E,XBOOLE_0:def 5;
             end;
           hence A misses B;
           end;
         suppose B = P^+;
           hence A misses B by E;
           end;
         end;
       end;
     end;
hence M is a_partition of the carrier of R1 by A,EQREL_1:def 4;
for a,b being Element of R st a in P^+ & b in P^+ holds a + b in P^+
  by IDEAL_1:def 1;
hence S^+ is add-closed mult-closed;
end;
