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

theorem Th14:
  -infty < x & x < +infty & x <> 0 implies |.x.|*|. 1. / x .| = 1.
proof
  assume that
A1: -infty < x & x < +infty and
A2: x <> 0;
  reconsider a = x as Element of REAL by A1,XXREAL_0:14;
A3: 1./x = 1/a by Th2;
  per cases;
  suppose 0 <= x;
    then |.x.| = a & |. 1./x .| = 1/a by A3,Def1;
    then |.x.|*|. 1./x .| = a * (1/a) by XXREAL_3:def 5;
    hence thesis by A2,XCMPLX_1:106;
  end;
  suppose
A4: not 0 <= x;
    then
A5: |.x.| = -x by Def1
      .= -a by SUPINF_2:2;
    1./x < 0 by A3,A4;
    then |. 1./x .| = -(1./x) by Def1
    .= -(1/a) by A3,SUPINF_2:2;
    then |.x.|*|. 1./x .| = (-a) * (-(1/a)) by A5,XXREAL_3:def 5
      .= a * (1/a);
    hence thesis by A4,XCMPLX_1:106;
  end;
end;
