Defining inertial mass scale

I am following the approach to define mass scale as given in the book 'Newtonian Mechanics' by A.P.French (page 171).

Take different objects labeled 1, 2 and 3, pull them one at a time with the same force and measure individual accelerations: $a_1,a_2,a_3$. Proportionality factors $m_1,m_2,m_3$ are unknown yet. Then to define an inertial mass scale, we use these experimental results by putting $F= m_1a_1=m_2a_2=m_3a_3$. From this, we get: $\dfrac{m_2}{m_1}=\dfrac{a_1}{a_2}, \dfrac{m_3}{m_1}=\dfrac{a_1}{a_3}$.

Now, choose $m_1$ arbitrary as 1 kg. A quantitative measure of other objects can be obtained as multiple of this standard object.

But to Prove the fact that the inertial mass obtained this way is truly a property of an object itself and independent of force used, following argument is used (book uses argument in words only, I have put symbols):

Repeating the above procedure in a similar way for a different force $F^{'}$, we get: $F^{'}= m^{'}_1a{'}_1=m{'}_2a{'}_2=m{'}_3a{'}_3$ and henceforth, the ratios: $\dfrac{m{'}_2}{m{'}_1}=\dfrac{a{'}_1}{a{'}_2}, \dfrac{m{'}_3}{m{'}_1}=\dfrac{a{'}_1}{a{'}_3}$. We find experimentally that $\dfrac{m{'}_2}{m{'}_1} = \dfrac{m_2}{m_1}, \dfrac{m{'}_3}{m{'}_1} = \dfrac{m_3}{m_1}$.

Now, from this, how to conclude that $m^{'}_1 = m_1$? (As only then, we can conclude the value of the mass does not change with the force used in the experiment).

Edit 1: After looking at the answer below, I have now doubt over my interpretation of the concluding remark in the book, quoting the remark: "The fact that these experimentally determined mass ratios are independent of the magnitude of the force establishes the inertial mass as a characteristic property of the object."

What would be the right interpretation of it?

Now, from this, how to conclude that $m^{'}_1 = m_1$? (As only then, we can conclude the value of the mass does not change with the force used in the experiment).
You applied a certain procedure using force $F$ to assign the three masses, and then you repeated the procedure using force $F'$. The procedure involves arbitrarily assigning a mass of 1 kg to $m_1$, so as written, this does not prove that $m_1=m_1'$. You've simply chosen $m_1$ as your 1 kg standard, and then made the same choice again.