StrmCkr wrote:something like this?

Discontinuous Nice Loop: 4/5 r3c8 =9= r4c8 -9- r5c7 =9= r5c2 -9- r2c2 =9= r1c3 -9- r1c7 =9= r3c8 => r3c8<>4, r3c8<>5

I think that is indeed an example of reusing a candidate with opposite truth values (the 9 in r3c8) but the unexpressive nice loop notation is hiding it. That's why I wouldn't do it like that. If you insist on using nice loop notation, I'd use the Type 2 Discontinuous X-Cycle (=Nice Loop) which proves that r3c8=9 (simpler than proving it's not 4 or 5, and also avoids the reuse issue):

Discontinuous X-Cycle Type 2: r3c8 =9= r4c8 -9- r5c7 =9= r5c2 -9- r2c2 =9= r1c3 -9- r1c7 =9= r3c8 => r3c8=9

That's definitely correct. I'd rather use Eureka anyway, and I can see two legal ways to do it:

1) X-Chain: (9)r3c8=r4c8-r5c7=r5c2-r2c2=r1c3-r1c7=(9)r3c8 => -45 r3c8

2) Discontinuous X-Cycle Type 2: (9)r3c8=r4c8-r5c7=r5c2-r2c2=r1c3-r1c7=(9)r3c8 => 9r3c8

(They're obviously the same exact chain, but the perspective is different.)

If you really want to use the eliminating discontinuous loop type, then it's not only more complicated but possibly incorrect because of the reuse (though it may not matter because it happens in one cell). Anyway, this is how I would notate it:

3.1) Discontinuous Nice Loop Type 1: ((4|5)-9)r3c8=r4c8-r5c7=r5c2-r2c2=r1c3-r1c7=(9-(4|5))r3c8 => -45 r3c8

or this way to see more clearly the reuse problem with (9)r3c8:

3.2) Discontinuous Nice Loop Type 1: (4|5)r3c8

-(9)r3c8=r4c8-r5c7=r5c2-r2c2=r1c3-r1c7

=(9)r3c8-(4|5)r3c8 => -45 r3c8

Regardless of the notation, the Discontinuous Nice Loop Type 1 is the most complicated, so why would we want to do it that way? I'd use Discontinuous Nice Loop (X-Cycle) Type 2 here, because it's simplest and avoids any conflicts (the only discontinuity is in the start/end cell, which is legal for sure). To be more exact, the discontinuity is only in the start/end cell in both loops, but there are two of them in the eliminating loop type. Not sure if that's a problem (not in practice here anyway, because it yields a correct result).