## 2 Preliminaries
For \(k=2\) we get:  
\[\alpha\oplus_{\mathbb{C}}\beta=\left\langle C_{\lambda}^{*}(\mu_{\alpha},\mu_{ \beta}),C(\
u_{\alpha},\
u_{\beta})\right\rangle=\left\langle n_{\lambda}( \varphi^{-1}[\varphi(n_{\lambda}(\mu_{\alpha}))+\varphi(n_{\lambda}(\mu_{ \beta}))]),\,\varphi^{-1}[\varphi(\
u_{\alpha})+\varphi(\
u_{\beta})]\right\rangle.\]9  
\[\alpha\otimes_{\mathbb{C}}\beta=\left\langle C(\mu_{\alpha},\mu_{\beta}),C_{ \lambda}^{*}(\
u_{\alpha},\
u_{\beta})\right\rangle=\left\langle\varphi^{-1}[ \varphi(\mu_{\alpha})+\varphi(\mu_{\beta})],\,n_{\lambda}(\varphi^{-1}[ \varphi(n_{\lambda}(\
u_{\alpha}))+\varphi(n_{\lambda}(\
u_{\beta}))])\right\rangle.\]0  
We assume that for \(k=n\) holds. Then for \(k=n+1\):  
\[\alpha\otimes_{\mathbb{C}}\beta=\left\langle C(\mu_{\alpha},\mu_{\beta}),C_{ \lambda}^{*}(\
u_{\alpha},\
u_{\beta})\right\rangle=\left\langle\varphi^{-1}[ \varphi(\mu_{\alpha})+\varphi(\mu_{\beta})],\,n_{\lambda}(\varphi^{-1}[ \varphi(n_{\lambda}(\
u_{\alpha}))+\varphi(n_{\lambda}(\
u_{\beta}))])\right\rangle.\]1  
\[\alpha\otimes_{\mathbb{C}}\beta=\left\langle C(\mu_{\alpha},\mu_{\beta}),C_{ \lambda}^{*}(\
u_{\alpha},\
u_{\beta})\right\rangle=\left\langle\varphi^{-1}[ \varphi(\mu_{\alpha})+\varphi(\mu_{\beta})],\,n_{\lambda}(\varphi^{-1}[ \varphi(n_{\lambda}(\
u_{\alpha}))+\varphi(n_{\lambda}(\
u_{\beta}))])\right\rangle.\]2  
\[\alpha\otimes_{\mathbb{C}}\beta=\left\langle C(\mu_{\alpha},\mu_{\beta}),C_{ \lambda}^{*}(\
u_{\alpha},\
u_{\beta})\right\rangle=\left\langle\varphi^{-1}[ \varphi(\mu_{\alpha})+\varphi(\mu_{\beta})],\,n_{\lambda}(\varphi^{-1}[ \varphi(n_{\lambda}(\
u_{\alpha}))+\varphi(n_{\lambda}(\
u_{\beta}))])\right\rangle.\]3  
\[\alpha\otimes_{\mathbb{C}}\beta=\left\langle C(\mu_{\alpha},\mu_{\beta}),C_{ \lambda}^{*}(\
u_{\alpha},\
u_{\beta})\right\rangle=\left\langle\varphi^{-1}[ \varphi(\mu_{\alpha})+\varphi(\mu_{\beta})],\,n_{\lambda}(\varphi^{-1}[ \varphi(n_{\lambda}(\
u_{\alpha}))+\varphi(n_{\lambda}(\
u_{\beta}))])\right\rangle.\]4  
Hence, by mathematical induction the proof is complete. The proofs that the rest of the operations belong to the set \(V\) are similar. Next we investigate the properties of the proposed operations: \(\Box\)  
**Theorem 2**. : _Let \(\alpha=\left\langle\mu_{\alpha},\
u_{\alpha}\right\rangle\in V,\beta=\left\langle \mu_{\beta},\
u_{\beta}\right\rangle\in V\) and real valued parameters \(k,k_{1},k_{2}\geq 0\), then:_  
1. \(\alpha\oplus_{C}\beta=\beta\oplus_{C}\alpha\)_._
2. \(\alpha\otimes_{C}\beta=\beta\otimes_{C}\alpha\)_._
3. \(k(\alpha\oplus_{C}\beta)=k\alpha\oplus_{C}k\beta\)_._
4. \((\alpha\otimes_{C}\beta)^{k}=\alpha^{k}\otimes_{C}\beta^{k}\)_._
5. \(k_{1}\alpha\oplus_{C}k_{2}\alpha=(k_{1}+k_{2})\alpha\)_._
6. \(\alpha^{k_{1}}\otimes_{C}\alpha^{k_{2}}=\alpha^{k_{1}+k_{2}}\)_._
7. \(n_{\lambda}(\alpha)\oplus_{C}n_{\lambda}(\beta)=n_{\lambda}(\alpha\otimes_{C}\beta)\)_._
8. \(n_{\lambda}(\alpha)\otimes_{C}n_{\lambda}(\beta)=n_{\lambda}(\alpha\otimes_{C}\beta)\)_._
9. \(kn_{\lambda}(\alpha)=n_{\lambda}(\alpha^{k})\)_._
10. \((n_{\lambda}(\alpha))^{k}=n_{\lambda}(k\alpha)\)_._  
**Proof of Theorem 2. ** The cases 1 and 2 are obvious, because of Archimedean copulas and the corresponding co-copulas commutativity.