Intuition behind the positive slope of LM curve : 'Income must increase'? Source: by John Floyd PhD  (in Economics; U Chicago)


  The intuition behind the positive slope of LM is as follows:
  An increase in the interest rate reduces the demand for money
  and an increase
  in income increases it [the demand for money].
  To [...] [equalize] the demand for money equal to a constant money supply
   as the interest rate rises
  and we move along the LM curve,
  the level of income must increase.


Would someone please explain the bolded? It sounds too imperious and presumptuous:
how MUST a rise in interest rate increase consumers' level of income?
Eg, employers can still refuse to increase salary. Then income would NOT increase?
The intertemporal consumption equation is :

$$Y_{1}+\frac{Y_{2}}{(1+r)} = W$$

where $W$ is total wealth and $Y$ is revenue.

When interest rate rise investiment drops and savings rise, so lets say someone choses to save all his revenu from period 1 (this is just to calculate his total wealth), then in period 2 his total wealth will be :

$$Y_{1}(1+r)+Y_{2} = W(1+r)$$

So as you can see his total wealth in 2nd period grows when interest rate grows 

Also we can isolate $Y_{1}$ : 

$$Y_{1}= W-\frac{Y_{2}}{(1+r)}$$

And you see that when interest rate rises your revenu in period one rises.

I am not sure if this is a valid answer but it seems correct