reserve x,y,w,z for ExtReal,
  a for Real;

theorem
  x <> 0 implies |. 1./x .| = 1./|.x.|
proof
  assume
A1: x <> 0;
  per cases;
  suppose
A2: x = +infty;
    then |. 1./x .| = 0 by Def1;
    hence thesis by A2,Def1;
  end;
  suppose
    x = -infty;
    then |. 1./x .| = 0 & |.x.| = +infty by Def1,XXREAL_3:5;
    hence thesis;
  end;
  suppose
A3: x <> +infty & x <> -infty;
A4: 0 < |.x.| by A1,Th4;
A5: x < +infty by A3,XXREAL_0:4;
    -infty <= x by XXREAL_0:5;
    then
A6: -infty < x by A3,XXREAL_0:1;
    then -(+infty) < x by XXREAL_3:def 3;
    then
A7: |.x.| < +infty by A5,Th11;
    |. 1./x .|*|.x.| = 1. by A1,A6,A5,Th14;
    hence thesis by A4,A7,XXREAL_3:88;
  end;
end;
