I posted an answer and tried to make some of my hints invisible (but visible when you mouse-over them).
Formatting didn't quite work, and I split the comment into three lines making only the first and the third invisible, as follows.
If $K=1-\frac{K_1}{2K_2}$ then clearly $K<1$.
Let $\lambda=\frac1{2K_2}$ so $K=1-\lambda K_1$.
Then $0\le\frac12=1-\frac12=1-\lambda K_2\le1-\lambda F'(c)\leq1-\lambda K_1=K<1$.
If I try to make all three lines invisible, this is what I get (and it is obviously not what I want).
! If $K=1-\frac{K_1}{2K_2}$ then clearly $K<1$.
! Let $\lambda=\frac1{2K_2}$ so $K=1-\lambda K_1$.
! Then $0\le\frac12=1-\frac12=1-\lambda K_2\le1-\lambda F'(c)\leq1-\lambda K_1=K<1$.
Or if I combine all three into one, again I don't get the result I want:
! If $K=1-\frac{K_1}{2K_2}$ then clearly $K<1$. Let $\lambda=\frac1{2K_2}$ so $K=1-\lambda K_1$.
Then $0\le\frac12=1-\frac12=1-\lambda K_2\le1-\lambda F'(c)\leq1-\lambda K_1=K<1$.
There is either something that doesn't properly work as intended, or I do not understand the proper way to format. The answer where all this occurred (just in case you would end up voting :) is at https://math.stackexchange.com/q/3406622
Thank you for your help!