
theorem
for O being Ordering of INT.Ring holds O = Positives(INT.Ring)
proof
let O be Ordering of INT.Ring;
defpred P[Nat] means $1 in O;
B: 1.INT.Ring in O & 0.INT.Ring in O by ord3;
0.(INT.Ring) = 0 & 1.(INT.Ring) = 1 by INT_3:def 3; then
IA: P[0] by ord3;
E: O + O c= O by defpc;
IS: now let k be Nat;
    assume C: P[k];
    then consider a being Element of INT.Ring such that D: a = k;
    a + 1.INT.Ring in {x + y where x,y is Element of INT.Ring :
         x in O & y in O} by D,C,B;
    then k + 1 in O + O by INT_3:def 3,D,IDEAL_1:def 19;
    hence P[k+1] by E;
    end;
I: for k being Nat holds P[k] from NAT_1:sch 2(IA,IS);
now let o be object;
  assume o in Positives(INT.Ring);
  then consider i being Element of INT such that A: o = i & 0 <= i;
  i is Element of NAT by A,INT_1:3;
  hence o in O by A,I;
  end;
then Positives(INT.Ring) c= O;
hence thesis by ordsub;
end;
