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

theorem
  not((x=-infty or x=+infty) & (y=-infty or y=+infty)) & y<>0 implies |.
  x/y.| = |.x.|/|.y.|
proof
  assume that
A1: not((x=-infty or x=+infty) & (y=-infty or y=+infty)) and
A2: y<>0;
  reconsider y as R_eal by XXREAL_0:def 1;
  per cases;
  suppose
A3: x = +infty;
A4: -infty < y by A1,A3,XXREAL_0:12,14;
      per cases by A2;
      suppose
A5:     0 < y;
        then x / y = +infty by A3,A1,XXREAL_3:83;
        then
A6:     |.x/y.| = +infty by Def1;
        |.y.| = y by A5,Def1;
        hence thesis by A3,A1,A5,A6,XXREAL_3:83;
      end;
      suppose
A7:     y < 0;
        then |.y.| = -y by Def1;
        then
A8:    |.y.| < +infty by A4,XXREAL_3:5,38;
        x / y = -infty by A3,A1,A7,XXREAL_3:85;
        then |.x/y.| = +infty by Def1,XXREAL_3:5;
        hence thesis by A8,A2,A3,Th4,XXREAL_3:83;
      end;
  end;
  suppose
A9: x = -infty;
A10: -infty < y by A1,A9,XXREAL_0:12,14;
      per cases by A2;
      suppose
A11:    0 < y;
        then
A12:     x / y = -infty by A9,A1,XXREAL_3:86;
A13:    |.x.| = +infty by A9,Def1,XXREAL_3:5;
        |.y.| = y by A11,Def1;
        hence thesis by A9,A1,A11,A12,A13,XXREAL_3:83;
      end;
      suppose
A14:    y < 0;
        then |.y.| = -y by Def1;
        then
A15:    |.y.| < +infty by A10,XXREAL_3:5,38;
A16:    x / y = +infty by A9,A1,A14,XXREAL_3:84;
        0 < |.y.| & |.x.| = +infty by A2,A9,Th4,Def1,XXREAL_3:5;
        hence thesis by A15,A16,XXREAL_3:83;
      end;
  end;
  suppose
    x <> +infty & x <> -infty;
    then reconsider a = x as Element of REAL by XXREAL_0:14;
      per cases;
      suppose
        y = +infty;
        then |.x/y.| = 0 & |.y.| = +infty by Def1;
        hence thesis;
      end;
      suppose
        y = -infty;
        then |.x/y.| = 0 & |.y.| = +infty by Def1,XXREAL_3:5;
        hence thesis;
      end;
      suppose
        y <> +infty & y <> -infty;
        then reconsider b = y as Element of REAL by XXREAL_0:14;
        |.x.| = |.a qua Complex.| & |.y.| = |.b qua Complex.| by Th1;
        then
A17:    |.x.|/|.y.| = |.a qua Complex.|/|.b qua Complex.| by Th2;
        x/y = a/b by Th2;
        then |.x/y.| = |.a/b qua Complex.| by Th1;
        hence thesis by A17,COMPLEX1:67;
      end;
  end;
end;
