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Similarly to Theorem 13, it can easily be seen that (x^ ) becomes a properly efficient solution for (P), in the above theorem, if (gamma _{i} >0), for all (i=1,2,ldots,k).

This means that ( x ¯ is a properly efficient solution in problem (MWDP).

Then y ¯ is a properly efficient solution in the considered multitime multiobjective variational problem (MVP).

Let (hat{y}in Y) be a properly efficient solution in sense Geoffrion definition.

Then, (i) Assume that (hat{y}) is a properly efficient element of (1).

Hence, from Theorem 4.11 of [12], (x^ ) is a properly efficient solution for problem (P).

Assume that (hat{y}) is a properly efficient element of (1).

Theorem 4.3 Let ( y ¯ be a properly efficient solution in problem (MWDP) and y ¯ ∈ Γ ( Ω t 0, t 1 ).

Hence, the condition (ii) of Theorem 4.10 is false, and is not a properly efficient solution of this problem.

We shall prove that ( x ¯ is a properly efficient solution in (MWDP) by the method of contradiction.

If also all the hypotheses of Theorem  5.1 are satisfied, then ( x ¯ is a properly efficient solution in (WDP).