let z be complex number ; :: thesis: ( Arg z in ].0 ,(PI / 2).[ iff ( Re z > 0 & Im z > 0 ) )
A1: Arg z < 2 * PI by Th52;
thus ( Arg z in ].0 ,(PI / 2).[ implies ( Re z > 0 & Im z > 0 ) ) :: thesis: ( Re z > 0 & Im z > 0 implies Arg z in ].0 ,(PI / 2).[ )

proof

assume A2: Arg z in ].0 ,(PI / 2).[ ; :: thesis: ( Re z > 0 & Im z > 0 )
then A3: Arg z > 0 by 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;
cos (Arg z) > 0 by A2, SIN_COS:86;
hence Re z > 0 by A4, COMPLEX1:28; :: thesis: Im z > 0
Arg z < PI / 2 by A2, XXREAL_1:4;
then Arg z < PI by Lm2, XXREAL_0:2;
then Arg z in ].0 ,PI .[ by A3, XXREAL_1:4;
then sin . (Arg z) > 0 by Th23;
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 ].0 ,(PI / 2).[
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 sin (Arg z) > 0 by A6, COMPLEX1:28;
then A8: sin . (Arg z) > 0 by SIN_COS:def 21;
cos (Arg z) > 0 by A5, A7, COMPLEX1:28;
then cos . (Arg z) > 0 by SIN_COS:def 23;
then A9: not Arg z in [.(PI / 2),((3 / 2) * PI ).] by Th30;
0 <= Arg z by Th52;
then A10: Arg z > 0 by A8, SIN_COS:33;
not Arg z in [.PI ,(2 * PI ).] by A8, Th26;
then ( PI / 2 > Arg z or ( PI > Arg z & (3 / 2) * PI < Arg z ) ) by A1, A9, XXREAL_1:1;
hence Arg z in ].0 ,(PI / 2).[ by A10, Lm5, XXREAL_0:2, XXREAL_1:4; :: thesis: verum