
theorem
for R being ordered domRing,
    P being Ordering of R
for a being Element of R holds a = signum(P,a) '*' abs(P,a)
proof
let R be ordered domRing, P be Ordering of R, a be Element of R;
H: P \/ -P = the carrier of R by REALALG1:def 15;
per cases;
suppose A: a in P \ {0.R}; then
  B: a in P by XBOOLE_0:def 5;
  thus signum(P,a) '*' abs(P,a)
     = 1 '*' abs(P,a) by A,defsgn
    .= 1 '*' a by B,REALALG2:def 11
    .= a by RING_3:60;
  end;
suppose A: a = 0.R;
  hence signum(P,a) '*' abs(P,a)
      = 0 '*' abs(P,a) by defsgn
     .= a by A,RING_3:59;
  end;
suppose A: not a in P \ {0.R} & a <> 0.R; then
  not a in P or a in {0.R} by XBOOLE_0:def 5; then
  a in -P by A,H,XBOOLE_0:def 3,TARSKI:def 1; then
  -a in --P; then
  B: a <=P, 0.R;
  thus signum(P,a) '*' abs(P,a)
     = (-1) '*' abs(P,a) by A,defsgn
    .= (-1) '*' (-a) by B,REALALG2:70
    .= -(-a) by RING_3:61
    .= a;
  end;
end;
