
theorem
for R being preordered non degenerated Ring,
    P being Preordering of R,
    a being Element of R
holds -abs(P,a) <=P, a & a <=P, abs(P,a) iff a is P-ordered
proof
let R be preordered non degenerated Ring, O be Preordering of R,
    a be Element of R;
hereby assume AS: -abs(O,a) <=O, a & a <=O, abs(O,a);
  per cases;
  suppose a in -O;
    hence a is O-ordered by XBOOLE_0:def 3;
    end;
  suppose a in O;
    hence a is O-ordered by XBOOLE_0:def 3;
    end;
  suppose not(a in O) & not(a in -O);
    then 1.R <=O, a & a <=O, -1.R by AS,defa;
    then F: 1.R <=O, - 1.R by c3;
    -1.R <O, 0.R by c20;
    then -1.R <O, 1.R by c20,avb6;
    hence a is O-ordered by F,c2;
    end;
  end;
assume G: a is O-ordered;
   Y: 0.R <=O, abs(O,a) by G,av0; then
   X: abs(O,a) in O & O + O c= O by REALALG1:def 14;
   per cases by G,XBOOLE_0:def 3;
   suppose A: a in O;
     then C: abs(O,a) = a by defa;
     then B: abs(O,a) - a = 0.R by RLVECT_1:15;
     a + abs(O,a) in O + O by C,A;
     hence -abs(O,a) <=O,a by X;
     thus a <=O, abs(O,a) by B,REALALG1:25;
     end;
   suppose A: a in -O;
     then abs(O,a) = -a by defa;
     then abs(O,a) - (-a) = 0.R by RLVECT_1:15;
     then abs(O,a) + a in O by REALALG1:25;
     hence -abs(O,a) <=O,a;
     -a in --O by A;
     then abs(O,a) + -a in O+O by Y;
     hence a <=O, abs(O,a) by X;
   end;
end;
