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

theorem
  sgn (x/y) = (sgn x)/(sgn y)
proof
  per cases;
  suppose
A1: y = 0;
    hence sgn (x/y) = sgn (x*0") by XCMPLX_0:def 9
      .= (sgn x)*0" by Def2
      .= (sgn x)/0 by XCMPLX_0:def 9
      .= (sgn x)/(sgn y) by A1,Def2;
  end;
  suppose
A2: y <> 0;
    x/y = (x/y) * 1 .= (x/y) * (y * (1/y)) by A2,XCMPLX_1:106
      .= ((x/y) * y) * (1/y)
      .= x * (1/y) by A2,XCMPLX_1:87;
    then sgn (x/y) = sgn x * sgn (1/y) by Th18
      .= ((sgn x)/1) * (1/(sgn y)) by Th21
      .= (sgn x * 1)/(1 * sgn y) by XCMPLX_1:76
      .= (sgn x)/(1 * sgn y);
    hence thesis;
  end;
end;
