
theorem
  for x, y being Complex st Arg x < PI & Arg y < PI holds Arg(x+y ) < PI
proof
  let z1,z2 be Complex;
  assume that
A1: Arg(z1)<PI and
A2: Arg(z2)<PI;
A3: |.z2.| = |.z2.|+0*<i>;
A4: |.z1.| = |.z1.|+0*<i>;
  per cases by COMPTRIG:34;
  suppose
A5: Arg(z1)=0;
    then z1= |.z1.| by Th13;
    then
A6: Im z1=0 by A4,COMPLEX1:12;
    per cases by COMPTRIG:34;
    suppose
      Arg(z2)=0;
      then z2= |.z2.| by Th13;
      then
A7:   z1+z2= |.z1.| + |.z2.| + 0*<i> by A5,Th13;
      0<= |.z1.| & 0<= |.z2.| by COMPLEX1:46;
      hence thesis by A7,COMPTRIG:5,35;
    end;
    suppose
      Arg(z2)>0;
      then Arg(z2) in ].0,PI.[ by A2,XXREAL_1:4;
      then
A8:   Im z2>0 by Th16;
      Im (z1+z2)=(Im z1)+(Im z2) by COMPLEX1:8;
      then Arg(z1+z2) in ].0,PI.[ by A6,A8,Th16;
      hence thesis by XXREAL_1:4;
    end;
  end;
  suppose
    Arg(z1)>0;
    then Arg(z1) in ].0,PI.[ by A1,XXREAL_1:4;
    then
A9: Im z1>0 by Th16;
    per cases by COMPTRIG:34;
    suppose
      Arg(z2)=0;
      then z2= |.z2.| by Th13;
      then
A10:  Im z2=0 by A3,COMPLEX1:12;
      Im (z1+z2)=(Im z1)+(Im z2) by COMPLEX1:8;
      then Arg(z1+z2) in ].0,PI.[ by A9,A10,Th16;
      hence thesis by XXREAL_1:4;
    end;
    suppose
      Arg(z2)>0;
      then Arg(z2) in ].0,PI.[ by A2,XXREAL_1:4;
      then
A11:  Im z2>0 by Th16;
      Im (z1+z2)=(Im z1)+(Im z2) by COMPLEX1:8;
      then Arg(z1+z2) in ].0,PI.[ by A9,A11,Th16;
      hence thesis by XXREAL_1:4;
    end;
  end;
end;
