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

theorem Th72:
  angle(x,y,z) = 0 implies Arg(x-y) = Arg(z-y) & angle(z,y,x)=0
proof
  assume
A1: angle(x,y,z) =0;
  now
    per cases;
    case
      Arg(z-y)-Arg(x-y)>=0;
      then Arg(z-y)-Arg(x-y)=0 by A1,Def4;
      hence thesis by Def4;
    end;
    case
A2:   Arg(z-y)-Arg(x-y)<0;
      then -(Arg(z-y)-Arg(x-y))>0;
      then
A3:   angle(z,y,x)=Arg(x-y)-Arg(z-y) by Def4;
      angle(x,y,z)=2*PI+(Arg(z-y)-Arg(x-y)) by A2,Def4;
      hence contradiction by A1,A3,Th68;
    end;
  end;
  hence thesis;
end;
