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The nut broke off and had a small red ring/gasket between these connectors. I'm an noob but can follow instructions. Please give suggestions. Thanks
Below there is a water valve 
which may be used as an alternative if the above (cupper?) pipe is inconvenient to use.
Thanks, I read what you wrote and made a few adjustments. Here is what I'm trying:
From left to right:
•Toilet tank connector. •G3/8" F connector to G1/2" M. •Fiber(?) Gasket. •1/2" (15×21mm) F threaded connector to 16mm multicouche compression connector. •The white pipe (=multicouche, 16mm). •Same connector as last. •Rubber gasket. •G1/2" threadded connector to 10mm compression fitting. •Cupper pipe 10mm water supply.
The cupper pipe changes width quite frequently, unfortunately. But I think here is a good spot.
Why a fiber gasket and not rubber, if I may ask? I have rubber in the compression fitting. This feels better? Idk. I hope to replace the multicouche compression fitting with a press compression instead, for durability. This multicouche type is popular in France.
I'm not familiar with the term "multicouche" but it seems like what we call a composite pipe so essentially aluminium pipe encased in plastic. I've never used compression fittings for that, only press fittings.
Few points that came to mind:
If I were you I'd probably ditch the multicouche section entirely and find one of these:
There are ones with both straight and angled heads as well as ones with internal and external threads as well as 12 - 15 - 18mm ends that you can use the same compression fittings as for copper pipes. Here you could replace the gasgets and everything between them with just a 1/2" to 1/2" hose. You may even find a 1/2" to 3/8" one that you could attach directly to the toilet tank.
Regarding the gaskets, rubber ones seal better against uneven surfaces but fibre can withstand more heat and pressure. Rubber ones are also easy to overtighten. Tighten those with hand only and then around half a spin with a wrench. Don't let it twist while tightening.