reserve x for Real;

theorem Th45:
  for z be Complex st Im z > 0 holds sin Arg z > 0
proof
  let z be Complex;
  Re z < 0 or Re z = 0 or Re z > 0;
  then
A1: Re z < 0 or Re z > 0 or z = (0+Im z*<i>) by COMPLEX1:13;
  assume Im z > 0;
  then Arg z in ].PI/2,PI.[ or Arg z in ].0,PI/2.[ or Arg z = PI/2 by A1,Th37
,Th41,Th42;
  then
A2: PI/2 < Arg z & Arg z < PI or 0 < Arg z & Arg z < PI/2 or Arg z = PI/2 by
XXREAL_1:4;
  then Arg z < PI by Lm2,XXREAL_0:2;
  then Arg z in ].0,PI.[ by A2,XXREAL_1:4;
  then sin.Arg z > 0 by Th7;
  hence thesis by SIN_COS:def 17;
end;
