let z be complex number ; :: thesis: ( Arg z in ].PI ,((3 / 2) * PI ).[ iff ( Re z < 0 & Im z < 0 ) )
thus ( Arg z in ].PI ,((3 / 2) * PI ).[ implies ( Re z < 0 & Im z < 0 ) ) :: thesis: ( Re z < 0 & Im z < 0 implies Arg z in ].PI ,((3 / 2) * PI ).[ )

proof

assume A1: Arg z in ].PI ,((3 / 2) * PI ).[ ; :: thesis: ( Re z < 0 & Im z < 0 )
then PI < Arg z by XXREAL_1:4;
then A2: PI / 2 < Arg z by Lm2, XXREAL_0:2;
A3: Arg z > PI by A1, XXREAL_1:4;
then z <> 0 by Def1;
then A4: ( z = (|.z.| * (cos (Arg z))) + ((|.z.| * (sin (Arg z))) * <i> ) & |.z.| > 0 ) by Def1, COMPLEX1:133;
Arg z < (3 / 2) * PI by A1, XXREAL_1:4;
then Arg z in ].(PI / 2),((3 / 2) * PI ).[ by A2, XXREAL_1:4;
then cos . (Arg z) < 0 by Th29;
then cos (Arg z) < 0 by SIN_COS:def 23;
hence Re z < 0 by A4, COMPLEX1:28; :: thesis: Im z < 0
Arg z < (3 / 2) * PI by A1, XXREAL_1:4;
then Arg z < 2 * PI by Lm6, XXREAL_0:2;
then Arg z in ].PI ,(2 * PI ).[ by A3, XXREAL_1:4;
then sin . (Arg z) < 0 by Th25;
then sin (Arg z) < 0 by SIN_COS:def 21;
hence Im z < 0 by A4, COMPLEX1:28; :: thesis: verum

end;

assume that
A5: Re z < 0 and
A6: Im z < 0 ; :: thesis: Arg z in ].PI ,((3 / 2) * PI ).[
z = (Re z) + ((Im z) * <i> ) by COMPLEX1:29;
then z <> 0 + (0 * <i> ) by A5, COMPLEX1:163;
then A7: ( |.z.| > 0 & z = (|.z.| * (cos (Arg z))) + ((|.z.| * (sin (Arg z))) * <i> ) ) by Def1, COMPLEX1:133;
then cos (Arg z) < 0 by A5, COMPLEX1:28;
then cos . (Arg z) < 0 by SIN_COS:def 23;
then A8: not Arg z in [.((3 / 2) * PI ),(2 * PI ).] by Th32;
sin (Arg z) < 0 by A6, A7, COMPLEX1:28;
then sin . (Arg z) < 0 by SIN_COS:def 21;
then A9: not Arg z in [.0 ,PI .] by Th24;
Arg z < 2 * PI by Th52;
then A10: Arg z < (3 / 2) * PI by A8, XXREAL_1:1;
0 <= Arg z by Th52;
then Arg z > PI by A9, XXREAL_1:1;
hence Arg z in ].PI ,((3 / 2) * PI ).[ by A10, XXREAL_1:4; :: thesis: verum