violence on the swings

just back from a long yet lovely walk to one of our local parks. 

miss poppet and i made a bee line for the swings, of course. 

and were met by a little miss who didn't wish to give up either of the 2 swings. 

after much toing and froing i popped my poppet in swing number 1, much to this little misses horror. 

who then preceded to poke the big stick she was holding at the swing and then once i started pushing swung the stick right next to my miss poppet and struck the swing seat. 

now when faced with other children's misbehaviors i really struggle. what is appropriate for me to say or not to say? 

my initial reaction was a very stern 'no', while her grandma smiled and laughed from up in the adjacent play equipment. she then finally came over took the stick from the little girl, pretended to then hit her in the neck with it and then threw it in the bushes. 

hhmmhh??