The marginal propensity to consume and the multiplier according to Keynes

In "The General Theory of Employment, Interest and Money", Chapter 10: The marginal propensity to consume and the multiplier, John Maynard Keynes says:

Our normal psychological law that, when the real income of the community increases or decreases, its consumption will increase or decrease but not so fast, can, therefore, be translated¾not, indeed, with absolute accuracy but subject to qualifications which are obvious and can easily be stated in a formally complete fashion into the propositions that $$\Delta C_w$$ and $$\Delta Y_w$$ have the same sign, but $$\Delta Y_w >\Delta C_w$$, where $$C_w$$ is the consumption in terms of wage-units. This is merely a repetition of the proposition already established in Chapter 3 above. Let us define, then, $$dC_w/dY_w$$ as the marginal propensity to consume.

This quantity is of considerable importance, because it tells us how the next increment of output will have to be divided between consumption and investment. For $$\Delta Y_w = \Delta C_w + \Delta I_w$$, where $$C_w$$ and $$I_w$$ are the increments of consumption and investment; so that we can write $$\Delta Y_w = k\Delta I_w$$, where $$1-1/k$$ is equal to the marginal propensity to consume.

Let us call $$k$$ the investment multiplier. It tells us that, when there is an increment of aggregate investment, income will increase by an amount which is k times the increment of investment.

I don't understood how Keynes make this step: "For $$\Delta Y_w = \Delta C_w + \Delta I_w$$, where $$C_w$$ and $$I_w$$ are the increments of consumption and investment; so that we can write $$\Delta Y_w = k\Delta I_w$$, where $$1-1/k$$ is equal to the marginal propensity to consume."

How is "$$\Delta Y_w = k\Delta I_w$$" inferred from "$$\Delta Y_w = \Delta C_w + \Delta I_w$$"?

Since

$$\Delta Y_w= \Delta C_w+ \Delta I_w$$

we have

$$1=\frac{\Delta C_w}{\Delta Y_w}+\frac{\Delta I_w}{\Delta Y_w}$$

or

$$\frac{\Delta I_w}{\Delta Y_w}=1-\frac{\Delta C_w}{\Delta Y_w}$$

Setting $$1-1/k=\frac{\Delta C_w}{\Delta Y_w}$$ (the marginal propensity to consume), we have

$$\frac{\Delta I_w}{\Delta Y_w}=\frac{1}{k}$$

so that

$$\Delta Y_w=k\Delta I_w.$$

Alternatively, substituting $$\Delta C_w=(1-1/k)\Delta Y_w$$ into

$$\Delta Y_w= \Delta C_w+ \Delta I_w$$

gives:

$$\Delta Y_w= \left(1-\frac{1}{k}\right)\Delta Y_w +\Delta I_w$$

or

$$0= -\frac{1}{k}\Delta Y_w +\Delta I_w$$

so that

$$\Delta Y_w=k\Delta I_w.$$

Your notation isn't the cleanest, and my notation is a little shaky, but here is the idea. It's just algebra resulting from plugging $$\partial C_W / \partial Y_W = 1-1/k \implies \mathbf{D} C_W = (1-1/k) \mathbf{D}Y_W$$ into the identity $$\mathbf{D}Y_W = \mathbf{D} C_W+\mathbf{D} I_W.$$ After simplifying, you will see that the two equations are equivalent.