 Solution

Let ${\tau }_B=\inf \{u\ge t:S_u\le B\}$. While the contract is alive, the down-and-out put pays $(K-S_T{)}^{+}{1}_{\{{\tau }_B>T\}}$ at maturity; a prior touch has already knocked it out.

The truncated terminal payoff can be represented by the ordinary European-price combinationAt maturity its three regions areIndeed 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.