= Solution
{c}
Let $\tau_B=\inf\{u\geq t:S_u\leq B\}$. While the contract is alive, the <down-and-out put> pays $(K-S_T)^+\mathbf1_{\{\tau_B>T\}}$ at maturity; a prior touch has already knocked it out.
The truncated terminal payoff can be represented by the ordinary European-price combination
$$
F(S,t):=C(S,t;K)-C(S,t;B)+(K-B)C_d(S,t;B).
$$
At maturity its three regions are
$$
\boxed{F(S,T)=(K-S)^+\mathbf1_{\{S\geq B\}}.}
$$
Indeed it is zero below $B$, equals $K-S$ between $B$ and $K$, and is zero above $K$. This supplies the terminal building block for reflection.
The zero-below-barrier property is a property of the terminal payoff. Before maturity, $F(S,t)$ need not vanish for $S<B$, because the ordinary European contracts can still finish above $B$. The killed <barrier option> price, rather than this unrestricted European combination, is zero after a touch.
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