\(
\text{Goal 1:} I \text{ is a set of non-bases of a mtaroid } M\ \\
\text{Assume the opposite: } \exists a \in A: \forall b \in B \backslash A \implies A - a + b \not \in \mathcal{B} \\
\implies |B \backslash A| \geq 2 \implies \text{we take distinct } c, d \in B \backslash A. \\
\text{By assumption of the opposite } A - a + c \not \in \mathcal{B} \text{, .i.e. } A - a + c \in I \\
\text{By assumption of the opposite } A - a + d \not \in \mathcal{B} \text{, .i.e. } A - a + d \in I \\
\text{But } |(A - a + c) \cap (A - a + d)| = |A-a| = r-1 \\
\implies (A - a + c) \sim (A - a + d)
\text{ in } J(n,r) \\
\implies \color{red}{\text{contradicts with "} I \text{ is stable".}} \\
\text{Goal 2: } M \text{ is paving.} \\\
\text{Assume the opposite, i.e. } \exists \text{dep. set } D: |D| < r \\
\implies \exists D' \supseteq D: |D'| = r-1 \text{, since any superset of }D \text{ is dep.} \\
\text{ Any superset of } D' \text{ is dep.} \\
\implies \exists I_1, I_2 \in I: |I_1 \cap I_2| = |D'| = r-1 \\
\implies I_1 \sim I_2 \implies \color{red}{\text{contradicts with } I \text{ is stable }} \\
\text{Goal 3: } M \text{ is sparse, i. e. } M^{*} \text{ is paving. } \\\
\text{Dual of } J(n,r) \text{ is exactly } J(n,n-r) \text{, so the proof is the same as for } M \text{ is paving}.
\)