
theorem
  for x being R_eal st x < 0 holds |. x .| = -x by Def1;

registration let x be R_eal;
 cluster |.x.| -> non negative;
 coherence
  proof per cases;
   suppose
A1:   0 <= x;
     then |. x .| = x by Def1;
    hence thesis by A1;
   end;
   suppose
A2:   x < 0;
     then |. x .| = -x by Def1;
    hence thesis by A2;
   end;
  end;
