let x be Real; :: thesis: ( x < 0 implies Arg (x * <i> ) = (3 / 2) * PI )
assume A1: x < 0 ; :: thesis: Arg (x * <i> ) = (3 / 2) * PI
A2: ( 0 <= Arg (0 + (x * <i> )) & Arg (0 + (x * <i> )) < 2 * PI ) by Th52;
A3: 0 + (x * <i> ) <> 0 by A1;
then A4: |.(0 + (x * <i> )).| <> 0 by COMPLEX1:131;
A5: 0 + (x * <i> ) = (|.(0 + (x * <i> )).| * (cos (Arg (0 + (x * <i> ))))) + ((|.(0 + (x * <i> )).| * (sin (Arg (0 + (x * <i> ))))) * <i> ) by A3, Def1;
then cos (Arg (0 + (x * <i> ))) = 0 by A4, COMPLEX1:163;
then ( Arg (0 + (x * <i> )) = (3 / 2) * PI or |.(0 + (x * <i> )).| * 1 = x ) by A2, A5, Th34, SIN_COS:82;
hence Arg (x * <i> ) = (3 / 2) * PI by A1, COMPLEX1:132; :: thesis: verum