Aβt−1	≤0	Start with the half-space in time t − 1
$\iff A{f}_{t-1}^{-1}{M}_{+}^{-1}{M}_{+}{f}_{t-1}{\beta }_{t-1}$	≤0	Substitute $AI=A\left(f_{t-1}^{-1}{M}_{+}^{-1}{M}_{+}{f}_{t-1}\right)$
⇔CM+ft−1βt−1	≤0	Use definition of C
⇔N+(ft−1βt−1)CM+ft−1βt−1	≤ (N+(ft−1βt−1))(0)	Multiply both sides by positive scalar
⇔CN+(ft−1βt−1)M+ft−1βt−1	≤0	Move scalar
⇔Cβt	≤0	Use definition of (βt|βt−1, zt−1 = SS+)