
theorem ineq1:
for R being preordered domRing,
    P being Preordering of R,
    a being P-ordered Element of R,
    p being P-ordered non P-negative Element of R
holds abs(P,a) <=P, p iff (-p <=P, a & a <=P, p)
proof
let R be preordered domRing, P be Preordering of R,
    a be P-ordered Element of R,
    p be P-ordered non P-negative Element of R;
H: not(p in -P \ {0.R}) & p in P \/ -P by defn,defppp;
then not(p in -P) or p in {0.R} by XBOOLE_0:def 5;
then As: not(p in -P) or p = 0.R by TARSKI:def 1;
then AS: a in (P \/ (-P)) & 0.R <=P, p
   by defppp,H,XBOOLE_0:def 3,REALALG1:25;
hereby assume A1: abs(P,a) <=P, p;
     per cases by AS,XBOOLE_0:def 3;
     suppose a in P;
       then A2: 0.R <=P, a;
       A3: a <=P, abs(P,a) by ineq2;
       -p <=P, 0.R by As,H,XBOOLE_0:def 3,REALALG1:25;
       hence -p <=P, a & a <=P, p by A3,A2,A1,c3;
       end;
     suppose a in -P;
       then -a in --P;
       then A3: --a <=P, 0.R;
       -abs(P,a) <=P, a by ineq2;
       then -a <= P, --abs(P,a) by c10a;
       then -a <=P, p by A1,c3;
       hence -p <=P, a & a <=P, p by A3,AS,c3,c10a;
       end;
     end;
assume A1: -p <=P, a & a <=P, p;
     per cases by AS,XBOOLE_0:def 3;
     suppose a in P;
       then 0.R <=P, a;
       hence abs(P,a) <=P, p by A1,av2;
       end;
     suppose a in -P;
       then -a in --P;
       then --a <=P, 0.R;
       then abs(P,a) = -a by av3;
       hence abs(P,a) <=P, p by A1,c10a;
       end;
end;
