I tried to proof if there is exist a complete language in R department. My main idea was to show that there is exist complete language in RE department and conclude from that that there is exist a complete language in R because R⊆RE.
I got stucked in this stage, hope someone can help me out.
Full definition for complete lang in R deparment: For given lang department R , lets say language L1 is complete in R if L1⊆R and for each L2∈R , L2≤L1
Completeness only makes sense with respect to some reducibility. For example, the phrase "$\mathsf{NP}$-complete" refers to polynomial-time many-one reductions, whereas when we say that the Halting Problem is a complete r.e. set the relevant notion is arbitrary-time many-one reductions. And there are even reducibility notions which aren't of "many-one flavor" at all - e.g. Turing reducibility, which while less relevant in complexity theory is the main reducibility notion in computability theory.
That said, your question does have a strong negative answer. According to every reducibility notion I know of, either every set in $\mathsf{REC}$ is complete for $\mathsf{REC}$ with respect to that reducibility or there is no complete set in $\mathsf{REC}$ for that reducibility. In particular, Turing reducibility (trivially) has the first behavior, and every resource-bounded reducibility (e.g. polytime $m$-reducibility) has the second (because we can always "pad out" a given language in a computable way to make it more complicated with respect to a resource-bounded reducibility notion).
Note that this is in contrast with the class $\mathsf{RE}$. While the same argument as for $\mathsf{REC}$ shows that there is no $\mathsf{RE}$-complete language with respect to any resource-bounded reducibility, the situation is nontrivial for (resource-unbounded) many-one reducibility: the Halting Problem is $\mathsf{RE}$-complete, but the degree structure of r.e. sets modulo many-one reducibility (or even the coarser notion of Turing reducibility) is extremely rich. In general, $\mathsf{RE}$ is much better behaved than $\mathsf{REC}$, and this is one of the key foundational observations of computability theory; for example, we can enumerate the $\mathsf{RE}$ languages in an $\mathsf{RE}$ way but we can't enumerate the $\mathsf{REC}$ languages in a $\mathsf{REC}$ (or even $\mathsf{RE}$) way.