let z be Complex; :: thesis: ( Re z < 0 implies cos (Arg z) < 0 )
( Im z < 0 or Im z = 0 or Im z > 0 ) ;
then A1: ( Im z < 0 or Im z > 0 or z = (Re z) + (0 * <i>) ) by COMPLEX1:13;
assume Re z < 0 ; :: thesis: cos (Arg z) < 0
then ( Arg z in ].(PI / 2),PI.[ or Arg z in ].PI,((3 / 2) * PI).[ or Arg z = PI ) by A1, Th36, Th42, Th43;
then ( ( PI / 2 < Arg z & Arg z < PI ) or ( PI < Arg z & Arg z < (3 / 2) * PI ) or Arg z = PI ) by XXREAL_1:4;
then ( PI / 2 < Arg z & Arg z < (3 / 2) * PI ) by Lm2, Lm5, XXREAL_0:2;
then Arg z in ].(PI / 2),((3 / 2) * PI).[ by XXREAL_1:4;
then cos . (Arg z) < 0 by Th13;
hence cos (Arg z) < 0 by SIN_COS:def 19; :: thesis: verum