Proposition 1

Proof 1 Since we do not have any loop, a non-empty connected component has at least two vertices.

Proposition 2

Proof 2 Since we do not have any circuit, a non-empty strongly connected component consists of a single vertex.

Proposition 3

Proof 3 Since we do not have any loop, a non-empty strongly connected component has at least two vertices.

Proposition 4

Proof 4 Since we do not have any loop, a non-empty connected component has at least two vertices.

Proposition 5

Proof 5 Since we do not have any circuit, a non-empty strongly connected component consists of a single vertex.

Proposition 6

Proof 6 Since we do not have any loop, a non-empty strongly connected component has at least two vertices.

Proposition 7

Proof 7 By definition of $\mathrm{𝚘𝚗𝚎}\_\mathrm{𝚜𝚞𝚌𝚌}$.

Proposition 8

Proof 8 By definition of $\mathrm{𝚗𝚘}\mathrm{𝚕𝚘𝚘𝚙}$, each connected component has at least two vertices.

Proposition 9

Proof 9 By definition of $\mathrm{𝚌𝚘𝚗𝚜𝚎𝚌𝚞𝚝𝚒𝚟𝚎}\_\mathrm{𝚕𝚘𝚘𝚙𝚜}\_\mathrm{𝚊𝚛𝚎}\_\mathrm{𝚌𝚘𝚗𝚗𝚎𝚌𝚝𝚎𝚍}$.

Proposition 10

Proof 10 By definition of $\mathrm{𝚗𝚘}\mathrm{𝚕𝚘𝚘𝚙}$, each strongly connected component has at least two vertices.

Proposition 11

Proof 11 Since we do not have any isolated vertex.

Proposition 12

Proof 12 Since we do not have any isolated vertex.

Proposition 13

Proof 13 By definition of $\mathrm{𝚘𝚗𝚎}\_\mathrm{𝚜𝚞𝚌𝚌}$.