study exercise on natural deduction

Question

How would you solve this? Its for my exam in an hour.

{((∃x α) ∨ (∃x β))} l- (∃x (α ∨ β)).

1. ((∃x α) ∨ (∃x β)) premise
2. (∃x α) assumption
3. α[u/x], u fresh

^^ my approach was to assume one side at a time and conclude the conclusion using ve

Answer 1

01. [∃xα]∨[∃xβ]       premise
  02. ∃xα             assumption
    03. α[u/x]        assumption
    04. α[u/x]∨β[u/x] ∨intro 03
    05. ∃x[α∨β]       ∃intro 04
  06. ∃x[α∨β]         ∃elim 02 03-05
07. [∃xα]→[∃x[α∨β]]   →intro 02-06
  08. ∃xβ             assumption
    09. β[u/x]        assumption
    10. α[u/x]∨β[u/x] ∨intro 09
    11. ∃x[α∨β]       ∃intro 10
  12. ∃x[α∨β]         ∃elim 08 09-11
13. [∃xβ]→[∃x[α∨β]]   →intro 08-12
14. ∃x[α∨β]           ∨elim 01 07 13