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

theorem Th69:
  angle(x,y,z)=angle(x-y,0,z-y)
proof
  now
    per cases;
    case
A1:   Arg((z-y)-0c)-Arg((x-y)-0c)>=0;
      then angle(x-y,0c,z-y)=Arg(z-y)-Arg(x-y) by Def4;
      hence angle(x-y,0c,z-y)=angle(x,y,z) by A1,Def4;
    end;
    case
A2:   Arg((z-y)-0c)-Arg((x-y)-0c)<0;
      then angle(x-y,0c,z-y)=2*PI+(Arg(z-y)-Arg(x-y)) by Def4;
      hence angle(x-y,0c,z-y)=angle(x,y,z) by A2,Def4;
    end;
  end;
  hence thesis;
end;
