*Proposition 2*: *For any two populations* \mathcal{P}*and* \mathcal{Q}*, the distance between the (multilocus) genetic structures* *P* *and* *Q* *at any L gene loci (L ≥* 1*) of equal degree of ploidy N ≥* 1 *is not less than the mean distance between the corresponding single-locus structures* *P*^{(l)}*and* *Q*^{(l)}, *respectively, that is*,

\frac{1}{L}{\displaystyle \sum _{l=1}^{L}\Delta ({P}^{(l)},{Q}^{(l)})}\le \Delta (P,Q)

*where the difference between genetic types is measured by the elementary genic difference d*.

The validity of Proposition 2 for *L* = 1 is obvious. For *L* ≥ 2, proof depends on four lemmata that apply the following notation: Let *s*(*P*, *Q*) be a shift transformation between the *L*-locus genotypic structures. Denote the various *L*-locus types as *G*_{
x
}or *G*_{
y
}, and write each type *G*_{
x
}as the "product" {G}_{x}^{(l)}{G}_{x}^{(\{1,\mathrm{...},L\}\backslash l)} of its projection {G}_{x}^{(l)} to the single-locus type at loci *l* = 1 and its projection {G}_{x}^{C(l)} to the complementary (*L* - 1)-locus type. Denote the single-locus types at locus *l* as {g}_{u}^{(l)} or {g}_{u}^{(l)} and the complementary types as {g}_{u}^{C(l)} or {g}_{v}^{C(l)}. Define

{\mu}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)}):={\displaystyle \sum _{\left\{x|{G}_{x}^{(l)}={g}_{u}^{(l)}\right\}}{\displaystyle \sum _{\left\{y|{G}_{y}^{(l)}={g}_{v}^{(l)}\right\}}s({G}_{x},{G}_{y})}}

as the marginal sum of all shifts that involve the type {g}_{u}^{(l)} at locus *l* in the source type *G*_{
x
}and {g}_{u}^{(l)} in the sink type *G*_{
y
}.

*Lemma 2* *The difference* {\displaystyle {\sum}_{v}{\mu}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})}-{\displaystyle {\sum}_{v}{\mu}_{l}({g}_{v}^{(l)},{g}_{u}^{(l)})}*between the marginal sums for any u equals the net shift* {\displaystyle {\sum}_{v}{s}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})}-{\displaystyle {\sum}_{v}{s}_{l}({g}_{v}^{(l)},{g}_{u}^{(l)})} *for any shift transformation s*_{
l
}*at the locus*.

*Proof*: For the *l*-th locus it holds that:

\begin{array}{l}{\displaystyle \sum _{v}{\mu}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})}\hfill \\ =\hfill & {\displaystyle \sum _{u,t,w}s({g}_{u}^{(l)}{g}_{t}^{C(l)},{g}_{v}^{(l)}{g}_{w}^{C(l)})}\hfill \\ =\hfill & {\displaystyle \sum _{t}\left[P({g}_{u}^{(l)}{g}_{t}^{C(l)})-\mathrm{min}\phantom{\rule{0.1em}{0ex}}\left\{P({g}_{u}^{(l)}{g}_{t}^{C(l)}),Q({g}_{u}^{(l)}{g}_{t}^{C(l)})\right\}\right]}\hfill \\ {\displaystyle \sum _{v}{\mu}_{l}({g}_{v}^{(l)},{g}_{u}^{(l)})}\hfill \\ =\hfill & {\displaystyle \sum _{v,t,w}s({g}_{v}^{(l)}{g}_{w}^{C(l)},{g}_{u}^{(l)}{g}_{t}^{C(l)})}\hfill \\ =\hfill & {\displaystyle \sum _{t}\left[Q({g}_{u}^{(l)}{g}_{t}^{C(l)})-\mathrm{min}\phantom{\rule{0.1em}{0ex}}\left\{P({g}_{u}^{(l)}{g}_{t}^{C(l)}),Q({g}_{u}^{(l)}{g}_{t}^{C(l)})\right\}\right]}\hfill \end{array}

Their difference equals:

{\displaystyle \sum _{t}P({g}_{u}^{(l)}{g}_{t}^{C(l)})}-{\displaystyle \sum _{t}Q({g}_{u}^{(l)}{g}_{t}^{C(l)})}=P({g}_{u}^{(l)})-Q({g}_{u}^{(l)})

The same difference results for any shift transformation *s*_{
l
}at a locus *l*, since:

\begin{array}{l}{\displaystyle \sum _{v}{s}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})}={P}^{(l)}({g}_{u}^{(l)})-\mathrm{min}\phantom{\rule{0.1em}{0ex}}\{{P}^{(l)}({g}_{u}^{(l)}),{Q}^{(l)}({g}_{u}^{(l)})\}\hfill \\ {\displaystyle \sum _{v}{s}_{l}({g}_{v}^{(l)},{g}_{u}^{(l)})}={Q}^{(l)}({g}_{u}^{(l)})-\mathrm{min}\phantom{\rule{0.1em}{0ex}}\{{P}^{(l)}({g}_{u}^{(l)}),{Q}^{(l)}({g}_{u}^{(l)})\}\hfill \end{array}

■

Even though marginal sums share this property with any shift transformation at the locus, the following lemma shows that marginal sums may not specify a shift transformation.

*Lemma 3*: *The marginal sums* {\mu}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)}) *of all types* {g}_{u}^{(l)}, {g}_{u}^{(l)} *at locus l may shift an amount that is in excess of the amount required of any shift transformation at the locus*.

*Proof*: The total amount shifted away from any type {g}_{u}^{(l)} at locus *l* equals

\begin{array}{l}{\displaystyle \sum _{v}{\mu}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})}\hfill \\ =\hfill & {P}^{(l)}({g}_{u}^{(l)})-{\displaystyle \sum _{t}\mathrm{min}\phantom{\rule{0.1em}{0ex}}\left\{P({g}_{u}^{(l)}{g}_{t}^{C(l)}),Q({g}_{u}^{(l)}{g}_{t}^{C(l)})\right\}}\hfill \\ \ge \hfill & {P}^{(l)}({g}_{u}^{(l)})-\mathrm{min}\phantom{\rule{0.1em}{0ex}}\left\{{\displaystyle \sum _{t}P({g}_{u}^{(l)}{g}_{t}^{C(l)}),{\displaystyle \sum _{t}Q({g}_{u}^{(l)}{g}_{t}^{C(l)})}}\right\}\hfill \\ =\hfill & {P}^{(l)}({g}_{u}^{(l)})-\mathrm{min}\phantom{\rule{0.1em}{0ex}}\left\{{P}^{(l)}({g}_{u}^{(l)}),{Q}^{(l)}({g}_{u}^{(l)})\right\}\hfill \end{array}

By the same reasoning, the amount received by {g}_{u}^{(l)} equals

{\displaystyle \sum _{v}{\mu}_{l}({g}_{v}^{(l)},{g}_{u}^{(l)})\ge {Q}^{(l)}({g}_{u}^{(l)})}-\mathrm{min}\phantom{\rule{0.1em}{0ex}}\left\{{P}^{(l)}({g}_{u}^{(l)}),{Q}^{(l)}({g}_{u}^{(l)})\right\}

These inequalities contradict the equality required of a shift transformation. ■

Lemma 3 shows that the marginal sums may shift too much, and it is easy to construct examples for which this is the case. Excess amounts must be due to the appearance of one or more single-locus types both in two-locus source types and in two-locus sink types. This makes them both sources and sinks in the marginal sums, in violation of the properties of a shift transformation. The three ways in which a type {g}_{a}^{(l)} can act as both a source and a sink are:

\begin{array}{l}\begin{array}{cc}Case\phantom{\rule{0.5em}{0ex}}1:& {\mu}_{l}({g}_{a}^{(l)},{g}_{a}^{(l)})>0\end{array}\hfill \\ \begin{array}{ccccc}Case\phantom{\rule{0.5em}{0ex}}2:& {\mu}_{l}({g}_{a}^{(l)},{g}_{b}^{(l)})>0& \text{and}& {\mu}_{l}({g}_{b}^{(l)},{g}_{a}^{(l)})>0& (a\ne b)\end{array}\hfill \\ \begin{array}{ccccccccc}Case\phantom{\rule{0.5em}{0ex}}3:& {\mu}_{l}({g}_{a}^{(l)},{g}_{b}^{(l)})>0& \text{and}& {\mu}_{l}({g}_{b}^{(l)},{g}_{c}^{(l)})>0& (a\ne b& \text{and}& b\ne c& \text{and}& a\ne c)\end{array}\hfill \end{array}

The following lemma shows how to eliminate all ambivalent source/sink relationships from the marginal sums without changing the net amount shifted, *i.e*., amount sent away as a source minus the amount received as a sink.

*Lemma 4*: *The marginal sums* {\mu}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)}) *of all types* {g}_{u}^{(l)}, {g}_{u}^{(l)} *at locus l can be used to construct a quasi-shift κ*_{
l
}(*P*^{(l)}, *Q*^{(l)}) *with the following three properties:*

\begin{array}{c}{\displaystyle \sum _{u,v}d({g}_{u}^{(l)},{g}_{v}^{(l)})\cdot {\mu}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})}\ge {\displaystyle \sum _{u,v}d({g}_{u}^{(l)},{g}_{v}^{(l)})\cdot {\kappa}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})}\\ {\displaystyle \sum _{v}{\kappa}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})}\cdot {\displaystyle \sum _{v}{\kappa}_{l}({g}_{v}^{(l)},{g}_{u}^{(l)})}=0\\ {\displaystyle \sum _{v}{\mu}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})}-{\displaystyle \sum _{v}{\mu}_{l}({g}_{v}^{(l)},{g}_{u}^{(l)})}={\displaystyle \sum _{v}{\kappa}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})}-{\displaystyle \sum _{v}{\kappa}_{l}({g}_{v}^{(l)},{g}_{u}^{(l)})}=P({g}_{u}^{(l)})-Q({g}_{u}^{(l)})\end{array}

*Proof by construction*: Consider the following algorithm:

*START:* Set {\kappa}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})\leftarrow {\mu}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)}) for all *u*, *v*.

*Step 1:* If {\kappa}_{l}({g}_{a}^{(l)},{g}_{a}^{(l)})>0 holds for a type {g}_{a}^{(l)}, set {\kappa}_{l}({g}_{a}^{(l)},{g}_{a}^{(l)})\leftarrow 0. Since d({g}_{a}^{(l)},{g}_{a}^{(l)})=0, this has no effect on the sum {\displaystyle {\sum}_{u,v}d({g}_{u}^{(l)},{g}_{v}^{(l)})\cdot {\mu}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})}. Repeat for an additional type fulfilling the condition. If none exist, go to Step 2.

*Step 2:* If {\kappa}_{l}({g}_{a}^{(l)},{g}_{b}^{(l)})>0 and {\kappa}_{l}({g}_{b}^{(l)},{g}_{a}^{(l)})>0 hold for *a* ≠*b*, set

\begin{array}{lll}{{\kappa}^{\prime}}_{l}({g}_{a}^{(l)},{g}_{b}^{(l)})\hfill & \leftarrow \hfill & {\kappa}_{l}({g}_{a}^{(l)},{g}_{b}^{(l)})-M\hfill \\ {{\kappa}^{\prime}}_{l}({g}_{b}^{(l)},{g}_{a}^{(l)})\hfill & \leftarrow \hfill & {\kappa}_{l}({g}_{b}^{(l)},{g}_{a}^{(l)})-M\hfill \end{array}

where M:=\mathrm{min}\phantom{\rule{0.1em}{0ex}}\left\{{\kappa}_{l}({g}_{a}^{(l)},{g}_{b}^{(l)}),{\kappa}_{l}({g}_{b}^{(l)},{g}_{a}^{(l)})\right\}. Because

\begin{array}{l}d({g}_{a}^{(l)},{g}_{b}^{(l)})\cdot {{\kappa}^{\prime}}_{l}({g}_{a}^{(l)},{g}_{b}^{(l)})+d({g}_{b}^{(l)},{g}_{a}^{(l)})\cdot {{\kappa}^{\prime}}_{l}({g}_{b}^{(l)},{g}_{a}^{(l)})\hfill \\ =\hfill & d({g}_{a}^{(l)},{g}_{b}^{(l)})\cdot \left[{\kappa}_{l}({g}_{a}^{(l)},{g}_{b}^{(l)})+{\kappa}_{l}({g}_{b}^{(l)},{g}_{a}^{(l)})-2\cdot M\right]\hfill \\ \le \hfill & d({g}_{a}^{(l)},{g}_{b}^{(l)})\cdot {\kappa}_{l}({g}_{a}^{(l)},{g}_{b}^{(l)})+d({g}_{b}^{(l)},{g}_{a}^{(l)})\cdot {\kappa}_{l}({g}_{b}^{(l)},{g}_{a}^{(l)})\hfill \end{array}

it follows that

{\displaystyle \sum _{u,v}d({g}_{u}^{(l)},{g}_{v}^{(l)})\cdot {\kappa}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})\ge {\displaystyle \sum _{u,v}d({g}_{u}^{(l)},{g}_{v}^{(l)})\cdot {{\kappa}^{\prime}}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})}}

Set

\begin{array}{lll}{\kappa}_{l}({g}_{a}^{(l)},{g}_{b}^{(l)})\hfill & \leftarrow \hfill & {{\kappa}^{\prime}}_{l}({g}_{a}^{(l)},{g}_{b}^{(l)})\hfill \\ {\kappa}_{l}({g}_{b}^{(l)},{g}_{a}^{(l)})\hfill & \leftarrow \hfill & {{\kappa}^{\prime}}_{l}({g}_{b}^{(l)},{g}_{a}^{(l)})\hfill \end{array}

Repeat for an additional pair of types that fulfill the condition. If none exist, go to Step 3.

*Step 3:* If {\kappa}_{l}({g}_{a}^{(l)},{g}_{b}^{(l)})>0 and {\kappa}_{l}({g}_{b}^{(l)},{g}_{c}^{(l)})>0 hold for three different indices *a*, *b*, *c,* subtract M:=\mathrm{min}\phantom{\rule{0.1em}{0ex}}\left\{{\kappa}_{l}({g}_{a}^{(l)},{g}_{b}^{(l)}),{\kappa}_{l}({g}_{b}^{(l)},{g}_{c}^{(l)}\right\} from both and add *M* to the "direct route" from {g}_{a}^{(l)} to {g}_{c}^{(l)}, *i.e*., set

\begin{array}{lll}{{\kappa}^{\prime}}_{l}({g}_{a}^{(l)},{g}_{b}^{(l)})\hfill & \leftarrow \hfill & {\kappa}_{l}({g}_{a}^{(l)},{g}_{b}^{(l)})-M\hfill \\ {{\kappa}^{\prime}}_{l}({g}_{b}^{(l)},{g}_{c}^{(l)})\hfill & \leftarrow \hfill & {\kappa}_{l}({g}_{b}^{(l)},{g}_{c}^{(l)})-M\hfill \\ {{\kappa}^{\prime}}_{l}({g}_{a}^{(l)},{g}_{c}^{(l)})\hfill & \leftarrow \hfill & {\kappa}_{l}({g}_{a}^{(l)},{g}_{c}^{(l)})+M\hfill \end{array}

Because *d* is a metric distance, implying

d({g}_{a}^{(l)},{g}_{b}^{(l)})+d({g}_{b}^{(l)},{g}_{c}^{(l)})\ge d({g}_{a}^{(l)},{g}_{c}^{(l)})

it holds that

\begin{array}{l}d({g}_{a}^{(l)},{g}_{b}^{(l)})\cdot {\kappa}_{l}({g}_{a}^{(l)},{g}_{b}^{(l)})+d({g}_{b}^{(l)},{g}_{c}^{(l)})\cdot {{\kappa}^{\prime}}_{l}({g}_{b}^{(l)},{g}_{c}^{(l)})+d({g}_{a}^{(l)},{g}_{c}^{(l)})\cdot {{\kappa}^{\prime}}_{l}({g}_{a}^{(l)},{g}_{c}^{(l)})\hfill \\ =\hfill & d({g}_{a}^{(l)},{g}_{b}^{(l)})\cdot {{\kappa}^{\prime}}_{l}({g}_{a}^{(l)},{g}_{b}^{(l)})+d({g}_{b}^{(l)},{g}_{c}^{(l)})\cdot {\kappa}_{l}({g}_{b}^{(l)},{g}_{c}^{(l)})+d({g}_{a}^{(l)},{g}_{c}^{(l)})\cdot {\kappa}_{l}({g}_{a}^{(l)},{g}_{c}^{(l)})\hfill \\ -M\cdot \left[d({g}_{a}^{(l)},{g}_{b}^{(l)})+d({g}_{b}^{(l)},{g}_{c}^{(l)})-d({g}_{a}^{(l)},{g}_{c}^{(l)})\right]\hfill \\ \le \hfill & d({g}_{a}^{(l)},{g}_{b}^{(l)})\cdot {\kappa}_{l}({g}_{a}^{(l)},{g}_{b}^{(l)})+d({g}_{b}^{(l)},{g}_{c}^{(l)})\cdot {\kappa}_{l}({g}_{b}^{(l)},{g}_{c}^{(l)})+d({g}_{a}^{(l)},{g}_{c}^{(l)})\cdot {\kappa}_{l}({g}_{a}^{(l)},{g}_{c}^{(l)})\hfill \end{array}

from which it follows that

{\displaystyle \sum _{u,v}d({g}_{u}^{(l)},{g}_{v}^{(l)})\cdot {\kappa}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})\ge {\displaystyle \sum _{u,v}d({g}_{u}^{(l)},{g}_{v}^{(l)})\cdot {{\kappa}^{\prime}}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})}}

Set

\begin{array}{lll}{\kappa}_{l}({g}_{a}^{(l)},{g}_{b}^{(l)})\hfill & \leftarrow \hfill & {{\kappa}^{\prime}}_{l}({g}_{a}^{(l)},{g}_{b}^{(l)})\hfill \\ {\kappa}_{l}({g}_{b}^{(l)},{g}_{c}^{(l)})\hfill & \leftarrow \hfill & {{\kappa}^{\prime}}_{l}({g}_{b}^{(l)},{g}_{c}^{(l)})\hfill \\ {\kappa}_{l}({g}_{a}^{(l)},{g}_{c}^{(l)})\hfill & \leftarrow \hfill & {{\kappa}^{\prime}}_{l}({g}_{a}^{(l)},{g}_{c}^{(l)})\hfill \end{array}

If {{\kappa}^{\prime}}_{l}({g}_{c}^{(l)},{g}_{a}^{(l)})>0, go to Step 2. Otherwise, repeat Step 3 for another triplet of types fulfilling the condition. If none exists, *STOP*.

At each step, {\displaystyle {\sum}_{u,v}d({g}_{u}^{(l)},{g}_{v}^{(l)}){\kappa}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})} decreases or remains constant, yielding

{\displaystyle \sum _{u,v}d({g}_{u}^{(l)},{g}_{v}^{(l)})\cdot {\mu}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})\ge {\displaystyle \sum _{u,v}d({g}_{u}^{(l)},{g}_{v}^{(l)})\cdot {\kappa}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})}}

After completion, either {\displaystyle {\sum}_{v}{\kappa}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})}=0 or {\displaystyle {\sum}_{v}{\kappa}_{l}({g}_{v}^{(l)},{g}_{u}^{(l)})}=0 or both hold for all *u*, meaning that no type is both a source and a sink. The net quasi-shift {\displaystyle {\sum}_{v}{\kappa}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})}-{\displaystyle {\sum}_{v}{\kappa}_{l}({g}_{v}^{(l)},{g}_{u}^{(l)})} for each *u* remains constant throughout the algorithm, equaling P({g}_{u}^{(l)})-Q({g}_{u}^{(l)}) by Lemma 2. Thus the quasi-shifts *κ*_{
l
}({g}_{u}^{(l)}, {g}_{u}^{(l)}) fulfill the properties, as claimed. ■

*Lemma 5*: *The quasi-shifts κ*_{
l
}({g}_{u}^{(l)}, {g}_{u}^{(l)}) *constructed in Lemma 4 specify a shift transformation s*_{
l
}(*P*^{(l)}, *Q*^{(l)}) *for locus l for which it holds that*

{\displaystyle \sum _{u,v}d({g}_{u}^{(l)},{g}_{v}^{(l)})\cdot {\mu}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})\ge {\Delta}_{{s}_{l}}}({P}^{(l)},{Q}^{(l)})

*Proof*: As proven in Lemma 4, for the quasi-shifts {\kappa}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)}) it holds that

{\displaystyle \sum _{v}{\kappa}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})}-{\displaystyle \sum _{v}{\kappa}_{l}({g}_{v}^{(l)},{g}_{u}^{(l)})}=P({g}_{u}^{(l)})-Q({g}_{u}^{(l)})

and either {\displaystyle {\sum}_{v}{\kappa}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})}=0 or {\displaystyle {\sum}_{v}{\kappa}_{l}({g}_{v}^{(l)},{g}_{u}^{(l)})}=0 or both. There are three cases:

\begin{array}{ll}\text{If}\hfill & {\displaystyle \sum _{v}{\kappa}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})>0}\hfill \\ \text{then}\hfill & {\displaystyle \sum _{v}{\kappa}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})=P({g}_{u}^{(l)})-Q({g}_{u}^{(l)})}\hfill \\ \text{If}\hfill & {\displaystyle \sum _{v}{\kappa}_{l}({g}_{v}^{(l)},{g}_{u}^{(l)})>0}\hfill \\ \text{then}\hfill & {\displaystyle \sum _{v}{\kappa}_{l}({g}_{v}^{(l)},{g}_{u}^{(l)})=Q({g}_{u}^{(l)})-P({g}_{u}^{(l)})}\hfill \\ \text{If}\hfill & {\displaystyle \sum _{v}{\kappa}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})={\displaystyle \sum _{v}{\kappa}_{l}({g}_{v}^{(l)},{g}_{u}^{(l)})=0}}\hfill \\ \text{then}\hfill & P({g}_{u}^{(l)})=Q({g}_{u}^{(l)})\hfill \end{array}

These three cases can be combined to the expression

{\displaystyle \sum _{v}{\kappa}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})=P({g}_{u}^{(l)})-\mathrm{min}\phantom{\rule{0.1em}{0ex}}\left\{P({g}_{u}^{(l)}),Q({g}_{u}^{(l)})\right\}}

Therefore, the quasi-shifts *κ*_{
l
}({g}_{u}^{(l)}, {g}_{u}^{(l)}) fulfill the definition of alpha shift transformation at locus *l*. Defining the shift {s}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)}):={\kappa}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)}) and denoting {\Delta}_{{s}_{l}}({P}^{(l)},{Q}^{(l)}):={\displaystyle {\sum}_{u,v}d({g}_{u}^{(l)},{g}_{v}^{(l)})\cdot {s}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})}, it follows from Lemma 4 that ■

{\displaystyle \sum _{u,v}d({g}_{u}^{(l)},{g}_{v}^{(l)})\cdot {\mu}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})={\Delta}_{{s}_{l}}}({P}^{(l)},{Q}^{(l)})

With the help of the lemmata, Proposition 2 can now be proven:

*Proof of Proposition 2*: Let *s*(*P*, *Q*) be a shift transformation between the two *L*-locus genotypic structures. Denoting the *L*-locus types as *G*_{
x
}or *G*_{
y
}, their projections to locus *l* as {G}_{x}^{(l)} or {G}_{y}^{(l)}, and the various single-locus types at locus *l* as {g}_{u}^{(l)} or {g}_{u}^{(l)}, it holds that

\begin{array}{lll}{\Delta}_{s}(P,Q)\hfill & =\hfill & {\displaystyle \sum _{x,y}d({G}_{x},{G}_{y})\cdot s({G}_{x},{G}_{y})}\hfill \\ =\hfill & {\displaystyle \sum _{x,y}\left[\frac{1}{L}{\displaystyle \sum _{l=1}^{L}d({G}_{x}^{(l)},{G}_{y}^{(l)})}\right]\cdot s({G}_{x},{G}_{y})}\hfill \\ =\hfill & \frac{1}{L}{\displaystyle \sum _{l=1}^{L}{\displaystyle \sum _{u,v}d({g}_{u}^{(l)},{g}_{v}^{(l)})\cdot {\mu}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})}}\hfill \\ \le \hfill & \frac{1}{L}{\displaystyle \sum _{l=1}^{L}{\Delta}_{{s}_{l}}({P}^{(l)},{Q}^{(l)})}\hfill \\ \le \hfill & \frac{1}{L}{\displaystyle \sum _{l=1}^{L}\Delta ({P}^{(l)},{Q}^{(l)})}\hfill \end{array}

where *s*_{
l
}(*P*^{(l)}, *Q*^{(l)}) is the shift transformation constructed in Lemma 4. Since the inequality holds in particular if *s*(*P*, *Q*) is a minimal shift transformation, it follows, as claimed, that ■

\Delta (P,Q)\ge \frac{1}{L}{\displaystyle \sum _{l=1}^{L}\Delta ({P}^{(l)},{Q}^{(l)})}

Equality holds in Proposition 2 whenever the marginal sums for each locus *l* = 1,...,*L* specify a minimal shift transformation, *i.e*., when {\displaystyle {\sum}_{u,v}d({g}_{u}^{(l)},{g}_{v}^{(l)})\cdot {\mu}_{l}({g}_{u}^{(l)},{g}_{v}^{(l)})=\Delta ({P}^{(l)},{Q}^{(l)})}..