reserve x, y, z, r, s, t for Real;

theorem Th21:
  1/(sgn x) = sgn (1/x)
proof
  per cases;
  suppose
A1: x = 0;
    hence 1/(sgn x) = 1/0 by Def2
      .= sgn (1/x) by A1,Def2;
  end;
  suppose
A2: x <> 0;
    then (sgn x * sgn (1/x)) * (1/(sgn x)) = 1 * (1/(sgn x)) by Th20;
    then sgn (1/x) * (sgn x * (1/(sgn x))) = 1/(sgn x);
    then sgn (1/x) * 1 = 1 /(sgn x) by A2,Th16,XCMPLX_1:106;
    hence thesis;
  end;
end;
