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the GF is riding an old trek, sized as an 18" (she is only 5'7")
but measuring the ett it seems to be 560 (not sure i measured it right) the bike looks to big but is there some kind of formula/ chart etc to give us an idea of starting points of a bike which may fit...
here you go : 😉
This is quite a comprehensive calculator:
http://vannicholas.com/framesizingsteps.aspx
No idea if the results are sensible though...
yours is too complicated simon 😉 (my posting is the easy chart 😉 (no offence like...lol 
Neither specs the toptube length though...
ah well its just a rough guide to give op the idea of sizes... 😉
this may help 😕
http://www.singletrackworld.com/forum/topic/fanny-bashers
simple way of doing it is to get her to place her elbow on the tip of the saddle and see if she can touch the top cap on the steerer tube with her middle finger. if she can then the top tube length is correct for her height. if she cant reach it then the frame is too big for her, but if she can reach beyond the top cap then she probably needs a longer stem or a larger frame. being the size you stated i doubt the frame will me too small for her but may be a tad on the larger size for her. i'm a similar height and i've always gone for frames sizes of med (17.5 inches) or smaller.