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Source. Section 5, printed p. 152 (published PDF). The source gives the constructions and rank formula briefly; their proofs are expanded here.

Statement. Restricting a finite matroid $M=(E,\mathcal F)$ to $E_0\subseteq E$ preserves the rank of all its subsets. For $D=E\setminus E_0$, choose any base $J$ of the restriction to $D$. The family

$$\mathcal F_0=\{I\subseteq E_0:I\cup J\in\mathcal F\}$$

is a matroid on $E_0$, independent of the chosen base $J$. Its rank is

$$r_0(A)=r(A\cup D)-r(D)\qquad(A\subseteq E_0). \tag{1}$$

This is contraction of $D$, followed by retaining the ground set $E_0$. The source calls it contraction to $E_0$.

Proof. For restriction, the independent subsets of $A\subseteq E_0$ are exactly the same as in $M$, so heredity, the rank axiom and the rank value are unchanged.

For fixed $J$, the family $\mathcal F_0$ contains the empty set and is hereditary. If $I,K\in\mathcal F_0$ and $|I|<|K|$, apply augmentation in $M$ to $I\cup J$ and $K\cup J$. Their size difference is the same, and an augmenting element belongs to $K\setminus I$, because both contain $J$. This proves augmentation in $\mathcal F_0$, so it is a matroid.

Take a maximal member $I$ of $\mathcal F_0$ inside $A$. Then $J\cup I$ is maximal independent in $D\cup A$. An element of $A\setminus I$ cannot be added by maximality in $\mathcal F_0$. An element of $D\setminus J$ cannot be added because $J$ is maximal independent in $D$, and the larger set would contain the dependent set obtained by adding it to $J$. Thus

$$r(D\cup A)=|J|+|I|=r(D)+r_0(A),$$

which proves (1). This expression does not depend on $J$. In any matroid a subset is independent exactly when its rank equals its size. Hence the rank formula also proves that $\mathcal F_0$ is independent of the chosen base. $\square$

In particular, if an independent seed $J$ is contracted, then $I\subseteq E\setminus J$ is independent in the contraction exactly when $I\cup J$ is independent in the original matroid. Subsequent deletion of other prescribed seeds does not change ranks on the remaining subsets. These are the interfaces used in Theorem 1d and Theorem 2d.