reserve a, b, c, d, x, y, z for Complex;
reserve r for Real;

theorem Th68:
  0 <= angle(x,y,z) & angle(x,y,z) < 2*PI
proof
  now
    per cases;
    case
A1:   Arg(z-y)-Arg(x-y)>=0;
      Arg(x-y)>=0 by COMPTRIG:34;
      then Arg(z-y)+0<=Arg(z-y)+Arg(x-y) by XREAL_1:7;
      then
A2:   Arg(z-y)<2*PI & Arg(z-y)-Arg(x-y)<=Arg(z-y) by COMPTRIG:34,XREAL_1:20;
      angle(x,y,z)=Arg(z-y)-Arg(x-y) by A1,Def4;
      hence thesis by A1,A2,XXREAL_0:2;
    end;
    case
A3:   Arg(z-y)-Arg(x-y)<0;
      Arg(z-y)>=0 by COMPTRIG:34;
      then Arg(x-y)+0<=Arg(x-y)+Arg(z-y) by XREAL_1:7;
      then Arg(x-y)<2*PI & Arg(x-y)-Arg(z-y)<=Arg(x-y) by COMPTRIG:34
,XREAL_1:20;
      then (Arg(x-y)-Arg(z-y))<2*PI by XXREAL_0:2;
      then
A4:   2*PI-(Arg(x-y)-Arg(z-y))>0 by XREAL_1:50;
      Arg(z-y)-Arg(x-y) + 2*PI < 0 + 2*PI by A3,XREAL_1:8;
      hence thesis by A3,A4,Def4;
    end;
  end;
  hence thesis;
end;
