## 2 The Norm Systems
### The Repeatability Norms
: _In order to simplify the description of the diagnosis granule model, it will be assumed in the further part of the paper that the equality holds for sequences of scaling functions: \(\{Scn\}=\{Scn^{\prime}\}\)._  
**Theorem 5**. : _For any natural number \(n>1\):_  
1. \(\textbf{n-ScIFN}\subseteq\textbf{(n+1)-ScIFN}\subseteq[0,1]\times[0,1]\)_,_
2. \(n\text{-ScIFN}\subseteq\textbf{(n+1)-ScIFN}\)_._  
The membership and non-membership values \(\langle\mu,v\rangle\in[0,1]\times[0,1]\) such that \(\mu+v>1\), are replaced with \(\langle Sc_{n}(\mu),Sc_{n}(v)\rangle\in\textbf{1-ScIFN}\), for some \(n\), when the power scale is entered. Then, the aggregation operations on the _IFS_ can be extended to the \(n\)-_ScIFN_. **Theorem 6**. : _In any system, \(\textbf{S}_{\textbf{Yager}}=\langle[0,1],\textbf{\bullet}_{i},\textbf{\bullet }_{s},\textbf{\bullet}_{p},\textbf{\bullet}_{l},0,1\rangle\), for any \(\langle\mu_{1},v_{1}\rangle,\langle\mu_{2},v_{2}\rangle\in\textbf{1-ScIFN}\) and numer \(\lambda\in[0,1]\), the conditions are satisfied:_  
1. \(\langle\mu_{1}\bullet_{s}\mu_{2},v_{1}\bullet_{t}v_{2}\rangle\in\textbf{1-ScIFN}\)_,_
2. \(\langle\mu_{1}\bullet_{t}\mu_{2},v_{1}\bullet_{s}v_{2}\rangle\in\textbf{1-ScIFN}\)_,_  
_and in some systems of \(\textbf{S}_{\textbf{Yager}}\), the below conditions are also satisfied:_  
1. \(\langle\lambda\bullet_{1}\mu_{1},\lambda\bullet_{p}v_{1}\rangle\in\textbf{1-ScIFN}\)_,_
2. \(\langle\lambda\bullet_{p}\mu_{1},\lambda\bullet_{t}v_{1}\rangle\in\textbf{1-ScIFN}\)_._  
**Proof. **  
1. \(\langle\mu_{1},v_{1}\rangle,\langle\mu_{2},v_{2}\rangle\in\textbf{1-ScIFN}\) iff \(\mu_{1}+v_{1}\leq 1,\mu_{2}+v_{2}\leq 1\) iff \(\mu_{1}\leq 1-v_{1},\mu_{2}\leq 1-v_{2}\). Hence and from the monotonicity of the s-norm and its determination by the t-norms: \(\mu_{1}\bullet_{s}\mu_{2}\leq(1-v_{1})\bullet_{s}(1-v_{2})=1-(1-(1-v_{1}) \bullet_{s}(1-v_{2})=1-v_{1}\bullet_{t}v_{2}\) iff \(\mu_{1}\bullet_{s}\mu_{2}+v_{1}\bullet_{t}v_{2}\leq 1\) iff \(\langle\mu_{1}\bullet_{s}\mu_{2},v_{1}\bullet_{t}v_{2}\rangle\in\textbf{1-ScIFN}\). 2. \(\langle\mu_{1},v_{1}\rangle,\langle\mu_{2},v_{2}\rangle\in\textbf{1-ScIFN}\) iff \(\langle\
u_{1},\mu_{1}\rangle,\langle\
u_{2},\mu_{2}\rangle\in\textbf{1-ScIFN}\). Hence and from the point 1 it follows that: \(\langle v_{1}\bullet_{s}v_{2},\mu_{1}\bullet_{t}\mu_{2}\rangle\in\textbf{1-ScIFN}\). Which is equivalent to \(\langle\mu_{1}\bullet_{t}\mu_{2},v_{1}\bullet_{s}v_{2}\rangle\in\textbf{1-ScIFN}\). 3. Let \(\mu\bullet_{s}v=\max\{\mu,v\},\mu\bullet_{t}v=\min\{\mu,v\}\), then \(f_{s}(x)=x,f_{t}(x)=1-x\) (see Example 1); \(\langle\mu_{1},v_{1}\rangle\in\textbf{1-ScIFN}\) iff \(\mu_{1}+v_{1}\leq 1\); \(\lambda\bullet_{t}\mu_{1}=f_{s}^{-1}[\lambda f_{s}(v_{1})]=\lambda\mu_{1}\) and \(\lambda\bullet_{p}v_{1}=f_{s}^{-1}[\lambda f_{s}(v_{1})]=1-\lambda(1-v_{1})\); \(\lambda\bullet_{t}\mu_{1}+\lambda\bullet_{p}v_{1}=\lambda\mu_{1}+1-\lambda(1- v_{1})=1-\lambda+\lambda(\mu_{1}+v_{1})\). Since \(\lambda(\mu_{1}+v_{1})\leq\lambda\), so \(0\leq 1-\lambda+\lambda(\mu_{1}+v_{1})\leq 1\). 4. Let \(\mu\bullet_{s}v=\max\{\mu,v\},\mu\bullet_{t}v=\min\{\mu,v\}\), then \(f_{s}(x)=x,f_{t}(x)=1-x\); \(\langle\mu_{1},v_{1}\rangle\in\textbf{1-ScIFN}\) iff \(\mu_{1}+v_{1}\leq 1\) iff \(\langle v_{1},\mu_{1}\rangle\in\textbf{1-ScIFN}\). Hence and from point 3, it follows that: \(\langle\lambda\bullet_{t}v_{1},\lambda\bullet_{p}\mu_{1}\rangle\in\textbf{1-ScIFN}\). Which is equivalent to \(\langle\lambda\bullet_{p}\mu_{1},\lambda\bullet_{t}v_{1}\rangle\in\textbf{1-ScIFN}\). \(\Box\)  
**Remark 3**. : _Adopting the triangular norm generators from Example 2:_  
\(\lambda\bullet_{l}\mu_{1}=\lambda^{1/p}\mu_{1},\lambda\bullet_{p}v_{1}=1- \lambda^{1/p}(1-v_{1})\)_, for \(\mu_{1}=1/2,v_{1}=7/8,\lambda=1/4\) and \(p=2\) we get \(\langle\lambda\bullet_{l}\mu_{1},\lambda\bullet_{p}v_{1}\rangle=\langle 1/8,15/16 \rangle\
otin\textbf{1-ScIFN}\) (because \(1<1/8+15/16\))._  
**Theorem 7**. : _Allow in the system \(\mathbf{S_{Yager}}\), conditions 1-4 of Theorem 6 apply.