let x be Real; :: thesis: ( x < 0 implies Arg x = PI )
A1: ( 0 <= Arg (x + (0 * <i>)) & Arg (x + (0 * <i>)) < 2 * PI ) by Th34;
assume A2: x < 0 ; :: thesis: Arg x = PI
then A3: x + (0 * <i>) = (|.(x + (0 * <i>)).| * (cos (Arg (x + (0 * <i>))))) + ((|.(x + (0 * <i>)).| * (sin (Arg (x + (0 * <i>))))) * <i>) by Def1;
|.(x + (0 * <i>)).| <> 0 by A2, COMPLEX1:45;
then sin (Arg (x + (0 * <i>))) = 0 by A3, COMPLEX1:77;
then ( Arg (x + (0 * <i>)) = PI or |.(x + (0 * <i>)).| * 1 = x ) by A1, A3, Th17, SIN_COS:31;
hence Arg x = PI by A2, COMPLEX1:46; :: thesis: verum