let z be Complex; :: thesis: ( Im z < 0 implies sin (Arg z) < 0 )
( Re z < 0 or Re z = 0 or Re z > 0 ) ;
then A1: ( Re z < 0 or Re z > 0 or z = 0 + ((Im z) * <i>) ) by COMPLEX1:13;
assume Im z < 0 ; :: thesis: sin (Arg z) < 0
then ( Arg z in ].PI,((3 / 2) * PI).[ or Arg z in ].((3 / 2) * PI),(2 * PI).[ or Arg z = (3 / 2) * PI ) by A1, Th38, Th43, Th44;
then ( ( PI < Arg z & Arg z < (3 / 2) * PI ) or ( (3 / 2) * PI < Arg z & Arg z < 2 * PI ) or Arg z = (3 / 2) * PI ) by XXREAL_1:4;
then ( PI < Arg z & Arg z < 2 * PI ) by Lm5, Lm6, XXREAL_0:2;
then Arg z in ].PI,(2 * PI).[ by XXREAL_1:4;
then sin . (Arg z) < 0 by Th9;
hence sin (Arg z) < 0 by SIN_COS:def 17; :: thesis: verum