Let {u}_{m} be an r.t. extension of v to K({x}_{m}) defined by a minimal pair ({a}_{m},{\delta}_{m})\in \overline{K}\times {G}_{\overline{v}} for m=1,\dots ,n, where [K({a}_{1},\dots ,{a}_{n}):K]={\prod}_{m=1}^{n}[K({a}_{m}):K] and let u be the r.t. extension of v to K({x}_{1},\dots ,{x}_{n}) defined as in (4). Let {u}_{m}^{\prime} be an r.t. extension of v to K({x}_{m}) defined by a minimal pair ({a}_{m}^{\prime},{\delta}_{m}^{\prime})\in \overline{K}\times {G}_{\overline{v}} for m=1,\dots ,n, where [K({a}_{1}^{\prime},\dots ,{a}_{n}^{\prime}):K]={\prod}_{m=1}^{n}[K({a}_{m}^{\prime}):K] and let {u}^{\prime} be the r.t. extension of v to K({x}_{1},\dots ,{x}_{n}) defined as in (4). A relation between such kind of r.t. extensions of v to K({x}_{1},\dots ,{x}_{n}) can be defined so that u\le {u}^{\prime} if and only if {u}_{m}\le {u}_{m}^{\prime} for m=1,\dots ,n. This is an order relation, and if u\le {u}^{\prime}, then for each polynomial F\in K[{x}_{1},\dots ,{x}_{n}], u(F)\le {u}^{\prime}(F) is satisfied. Because, for F={\sum}_{{t}_{1},\dots ,{t}_{n}}{d}_{{t}_{1}\cdots {t}_{n}}{x}_{1}^{{t}_{1}}\cdots {x}_{n}^{{t}_{n}}\in K[{x}_{1},\dots ,{x}_{n}],
\begin{array}{rcl}u(F)& =& \underset{{t}_{1},\dots ,{t}_{n}}{inf}(v({d}_{{t}_{1}\cdots {t}_{n}})+{t}_{1}{u}_{1}({x}_{1})+\cdots +{t}_{n}{u}_{n}({x}_{n}))\\ \le & \underset{{t}_{1},\dots ,{t}_{n}}{inf}(v({d}_{{t}_{1}\cdots {t}_{n}})+{t}_{1}{u}_{1}^{\prime}({x}_{1})+\cdots +{t}_{n}{u}_{n}^{\prime}({x}_{n}))={u}^{\prime}(F).\end{array}
Now, let I be a wellordered set without the last element and {({w}_{i})}_{i\in I} be an ordered system of r.t. extensions of v to K({x}_{1},\dots ,{x}_{n}), where {w}_{i} is defined as in (4), i.e., {w}_{i} is the common extension of {w}_{{i}_{m}}, where {w}_{{i}_{m}} is the r.t. extension of v to K({x}_{m}) defined by the minimal pair ({a}_{{i}_{m}},{\delta}_{{i}_{m}})\in \overline{K}\times {G}_{\overline{v}}, where [K({a}_{{i}_{1}},\dots ,{a}_{{i}_{n}}):K]={\prod}_{m=1}^{n}[K({a}_{{i}_{m}}):K] for all i\in I. If {w}_{i}\le {w}_{j} for all i<j, then {({w}_{i})}_{i\in I} is an ordered system of r.t. extensions of v to K({x}_{1},\dots ,{x}_{n}). Then the valuation w of K({x}_{1},\dots ,{x}_{n}) defined as
w(F)=\underset{i}{sup}({w}_{i}(F))
(5)
for all F\in K[{x}_{1},\dots ,{x}_{n}] is an extension of v to K({x}_{1},\dots ,{x}_{n}) and it is called a limit of the ordered system {({w}_{i})}_{i\in I}.
If {w}_{m} is the restriction of w to K({x}_{m}) for m=1,\dots ,n, then {w}_{m} is the limit of the ordered system {({w}_{{i}_{m}})}_{i\in I} of r.t. extensions of v to K({x}_{m}). Also, w is the common extension of {w}_{1},\dots ,{w}_{n} to K({x}_{1},\dots ,{x}_{n}). Since {w}_{m} may not be an r.t. extension of v to K({x}_{m}), then w may not be an r.t. extension of v to K({x}_{1},\dots ,{x}_{n}).
If w={sup}_{i}{w}_{i} is a residual algebraic torsion extension of v to K({x}_{1},\dots ,{x}_{n}), then {G}_{v}\subseteq {G}_{w}\subseteq {G}_{\overline{v}} is satisfied. Some other properties of w are studied below.
Denote the extension of {w}_{i} to \overline{K}({x}_{1},\dots ,{x}_{n}) by {\overline{w}}_{i} and the extension of {w}_{{i}_{m}} to \overline{K}({x}_{m}) by {\overline{w}}_{{i}_{m}} for m=1,\dots ,n and for all i\in I.
Theorem 3.1 Let {({\overline{w}}_{i})}_{i\in I} be an ordered system of r.t. extensions of \overline{v} to \overline{K}({x}_{1},\dots ,{x}_{n}), where {\overline{w}}_{i} is defined as in (4), i.e., {w}_{i} is the r.t. extension of v to \overline{K}({x}_{1},\dots ,{x}_{n}) which is the common extension of {w}_{{i}_{m}} for m=1,\dots ,n and for all i\in I. Denote the restriction of {\overline{w}}_{i} to K({x}_{1},\dots ,{x}_{n}) by {w}_{i} and the restriction of \overline{v} to K({a}_{{i}_{1}},\dots ,{a}_{{i}_{n}}) by {v}_{i}={v}_{{a}_{{i}_{1}}\cdots {a}_{{i}_{n}}}. Then

1.
For all i,j\in I, i<j, one has {w}_{i}<{w}_{j}, i.e., {({w}_{i})}_{i\in I} is an ordered system of r.t. extensions of v to K({x}_{1},\dots ,{x}_{n}).

2.
For all i,j\in I, i<j, one has {k}_{{v}_{i}}\subseteq {k}_{{v}_{j}} and {G}_{{v}_{i}}\subseteq {G}_{{v}_{j}}.

3.
Suppose that \overline{w}={sup}_{i}{\overline{w}}_{i} and \overline{w} is not an r.t. extension of \overline{v} to \overline{K}({x}_{1},\dots ,{x}_{n}) and denote that w is the restriction of \overline{w} to K({x}_{1},\dots ,{x}_{n}). Then w={sup}_{i}{w}_{i} and {k}_{w}={\bigcup}_{i}{k}_{{v}_{i}} and {G}_{w}={\bigcup}_{i}{G}_{{v}_{i}}.
Proof For every i\in I and m=1,\dots ,n, denote that {f}_{{i}_{m}}=Irr({a}_{{i}_{m}},K) and {deg}_{{x}_{m}}{f}_{{i}_{m}}={n}_{{i}_{m}}.

1.
Since {\overline{w}}_{i}<{\overline{w}}_{j} for all i,j\in I, i<j, we have {w}_{i}\le {w}_{j}. We show that {w}_{i}<{w}_{j}. Assume that {w}_{i}={w}_{j}. Since {w}_{i} is the common extension of {w}_{{i}_{m}} and {w}_{j} is the common extension of {w}_{{j}_{m}} for m=1,\dots ,n, we have {w}_{{i}_{m}}={w}_{{j}_{m}} for m=1,\dots ,n. Since ({a}_{{i}_{m}},{\delta}_{{i}_{m}}) is a minimal pair of the definition of {w}_{{i}_{m}}, we have {\delta}_{{i}_{m}}={\delta}_{{j}_{m}} for m=1,\dots ,n. But it is a contradiction, because {({\overline{w}}_{{i}_{m}})}_{i\in I} is an ordered system of r.t. extensions of \overline{v} to \overline{K}({x}_{m}) and so {\overline{w}}_{{i}_{m}}<{\overline{w}}_{{j}_{m}}, i.e., {\delta}_{{i}_{m}}<{\delta}_{{j}_{m}} [3]. Hence {w}_{{i}_{m}}<{w}_{{j}_{m}} for m=1,\dots ,n. Since {w}_{i} and {w}_{j} are common extensions of {w}_{{i}_{m}} and {w}_{{j}_{m}} respectively for m=1,\dots ,n and for all i\in I, it is concluded that {w}_{i}<{w}_{j} for all i<j.

2.
It is enough to study for B=F({a}_{{i}_{1}},\dots ,{a}_{{i}_{n}})\in K[{a}_{{i}_{1}},\dots ,{a}_{{i}_{n}}], where F({x}_{1},\dots ,{x}_{n})\in K[{x}_{1},\dots ,{x}_{n}], {deg}_{{x}_{m}}F({x}_{1},\dots ,{x}_{n})<{n}_{{i}_{m}}.
It is seen that {v}_{i}(B)={v}_{i}(F({a}_{{i}_{1}},\dots ,{a}_{{i}_{n}}))=\overline{v}(F({a}_{{i}_{1}},\dots ,{a}_{{i}_{n}}))={v}_{j}(F({a}_{{j}_{1}},\dots ,{a}_{{j}_{n}}))={v}_{j}(B) by using the [[3], Th. 2.3] and this gives {G}_{{v}_{i}}\subseteq {G}_{{v}_{j}}.
Assume that {v}_{i}(B)=0. Then {v}_{j}(B)=0. Since ({a}_{{i}_{m}},{\delta}_{{i}_{m}}) is a minimal pair of the definition of {w}_{{i}_{m}} for m=1,\dots ,n, we have {B}^{\ast}=F{({a}_{{i}_{1}},\dots ,{a}_{{i}_{m1}},{x}_{m},{a}_{{i}_{m+1}},\dots ,{a}_{{i}_{n}})}^{\ast}=F{({a}_{{i}_{1}},\dots ,{a}_{{i}_{m1}},{a}_{{i}_{m}},{a}_{{i}_{m+1}},\dots ,{a}_{{i}_{n}})}^{\ast}\in {k}_{{v}_{i}} coincides with the F{({a}_{{j}_{1}},\dots ,{a}_{{j}_{m1}},{a}_{{j}_{m}},{a}_{{j}_{m+1}},\dots ,{a}_{{j}_{n}})}^{\ast} which is the residue of B in {k}_{{v}_{j}}. Hence {k}_{{v}_{i}}\subseteq {k}_{{v}_{j}} for all i,j\in I, i<j.

3.
Since \overline{w}={sup}_{i}{\overline{w}}_{i}, we have w={sup}_{i}{w}_{i} and w is not an r.t. extension of v to K({x}_{1},\dots ,{x}_{n}). Using [[3], Th. 2.3] and the definition of {w}_{{i}_{m}}, the proof can be completed. Take F({x}_{1},\dots ,{x}_{n})\in K[{x}_{1},\dots ,{x}_{n}] such that {deg}_{{x}_{m}}F<{n}_{{i}_{m}}. Since ({a}_{{i}_{m}},{\delta}_{{i}_{m}}) is a minimal pair of the definition of {w}_{{i}_{m}}, \overline{w}(F({x}_{1},\dots ,{x}_{n}))=w(F({x}_{1},\dots ,{x}_{n}))=\overline{v}(F({a}_{{i}_{1}},\dots ,{a}_{{i}_{n}}))={v}_{i}(F({a}_{{i}_{1}},\dots ,{a}_{{i}_{n}})). This means that {G}_{{v}_{i}}\subseteq {G}_{w} for all i\in I and so {\bigcup}_{i}{G}_{{v}_{i}}\subseteq {G}_{w}.
Conversely, let {v}_{i}^{m} be the restriction of {\overline{w}}_{i} to K({a}_{{i}_{1}},\dots ,{a}_{{i}_{m1}},{x}_{m},{a}_{{i}_{m+1}},\dots ,{a}_{{i}_{n}}) for m=1,\dots ,n and for all i\in I. Since {v}_{i}^{m}(P({a}_{{i}_{1}},\dots ,{a}_{{i}_{m1}},{x}_{m},{a}_{{i}_{m+1}},\dots ,{a}_{{i}_{n}}))={v}_{i}(P({a}_{{i}_{1}},\dots ,{a}_{{i}_{m}},\dots ,{a}_{{i}_{n}})), then w(P({x}_{1},\dots ,{x}_{n}))={v}_{i}(P({a}_{{i}_{1}},\dots ,{a}_{{i}_{n}}))\in {G}_{{v}_{i}} for every P({x}_{1},\dots ,{x}_{n})\in K[{x}_{1},\dots ,{x}_{n}]. This gives {G}_{w}\subseteq {\bigcup}_{i}{G}_{{v}_{i}}.
Now, assume that \overline{v}(F({a}_{{i}_{1}},\dots ,{a}_{{i}_{n}}))={v}_{i}(F({a}_{{i}_{1}},\dots ,{a}_{{i}_{m}}))=0. Then w(F({x}_{1},\dots ,{x}_{n}))=0 and since {deg}_{{x}_{m}}F({x}_{1},\dots ,{x}_{n})<{n}_{{i}_{m}}, F{({a}_{1},\dots ,{a}_{n})}^{\ast}, which is {v}_{i}residue of F({x}_{1},\dots ,{x}_{n}), coincides with the residue of F({x}_{1},\dots ,{x}_{n}) in {k}_{w}. This shows {k}_{{v}_{i}}\subseteq {k}_{w} for all i\in I, and then {\bigcup}_{i}{k}_{{v}_{i}}\subseteq {k}_{w} .
For the reverse inclusion, let P({x}_{1},\dots ,{x}_{n})\in K[{x}_{1},\dots ,{x}_{n}] and w(P({x}_{1},\dots ,{x}_{n}))=0. For m=1,\dots ,n, P{({a}_{{i}_{1}},\dots ,{a}_{{i}_{m1}},{x}_{m},{a}_{{i}_{m+1}},\dots ,{a}_{{i}_{n}})}^{\ast} is equal to P{({a}_{{i}_{1}},\dots ,{a}_{{i}_{m1}},{a}_{{i}_{m}},{a}_{{i}_{m+1}},\dots ,{a}_{{i}_{n}})}^{\ast}\in {k}_{{v}_{i}} and so P{({x}_{1},\dots ,{x}_{n})}^{\ast}=P{({a}_{{i}_{1}},\dots ,{a}_{{i}_{n}})}^{\ast}\in {k}_{{v}_{i}}. Hence {k}_{w}\subseteq {\bigcup}_{i}{k}_{{v}_{i}}.
□
The following theorem can be obtained as a result of Theorem 3.1.
Corollary 3.2 Under the above notations, let w be an r.a.t. extension of v to K({x}_{1},\dots ,{x}_{n}). Then the following are satisfied:

1.
{G}_{{v}_{i}}\subseteq {G}_{{v}_{j}} and {k}_{{v}_{i}}\subseteq {k}_{{v}_{j}} for all i,j\in I, i<j.

2.
{({w}_{i})}_{i\in I} is an ordered system of r.t. extensions of v to K({x}_{1},\dots ,{x}_{n}) and w={sup}_{i}{w}_{i}. Moreover, we have {k}_{w}={\bigcup}_{i}{k}_{{v}_{i}} and {G}_{w}={\bigcup}_{i}{G}_{{v}_{i}}.
Proof If w is an r.a.t. extension of v to K({x}_{1},\dots ,{x}_{n}), then \overline{w} is an r.a.t. extension \overline{v} to \overline{K}({x}_{1},\dots ,{x}_{n}) and so {\overline{w}}_{m} is an r.a.t. extension of \overline{v} to \overline{K}({x}_{m}) for m=1,\dots ,n. We can take {\{{\delta}_{{i}_{m}}\}}_{i\in I} for m=1,\dots ,n as cofinal wellordered subsets of {M}_{{\overline{w}}_{m}}=\{\overline{w}({x}_{m}a)a\in \overline{K}\}. I has no last element because {\overline{w}}_{m} is not an r.t. extension of \overline{v}. For every i\in I, choose the element ({a}_{{i}_{m}},{\delta}_{{i}_{m}})\in \overline{K}\times {G}_{\overline{v}} such that for m=1,\dots ,n, \overline{w}({x}_{m}{a}_{{i}_{m}})={\delta}_{{i}_{m}} and [K({a}_{{i}_{m}}):K] is the smallest possible for {\delta}_{{i}_{m}}. This means that if {c}_{m}\in \overline{K} such that \overline{w}({x}_{m}{c}_{m})={\delta}_{{i}_{m}}, then [K({c}_{m}):K]\ge [K({a}_{{i}_{m}}):K]. Then ({a}_{{i}_{m}},{\delta}_{{i}_{m}}) is a minimal pair of the definition of {\overline{w}}_{{i}_{m}} with respect to K for m=1,\dots ,n. According to [[3], Th. 4.1], {\overline{w}}_{{i}_{m}}<{\overline{w}}_{{j}_{m}} if i<j, which means that {({\overline{w}}_{{i}_{m}})}_{i\in I} is an ordered system of r.t. extensions of \overline{v} to \overline{K}({x}_{m}) for m=1,\dots ,n and {({\overline{w}}_{{i}_{m}})}_{i\in I} has a limit {\overline{w}}_{m}={sup}_{i}{\overline{w}}_{{i}_{m}} which is an r.a.t extension of v to \overline{K}({x}_{m}). For all i\in I, take {\overline{w}}_{i} as the common extension of {\overline{w}}_{{i}_{m}} to K({x}_{1},\dots ,{x}_{n}) and \overline{w} as the common extension of {\overline{w}}_{m} to \overline{K}({x}_{1},\dots ,{x}_{n}) for m=1,\dots ,n. Denote the restriction of {\overline{w}}_{i} to K({x}_{1},\dots ,{x}_{n}) by {w}_{i} and denote the restriction of \overline{w} to K({x}_{1},\dots ,{x}_{n}) by w. In the same way as that in the proof of Theorem 3.1, it is seen that {w}_{i}<{w}_{j} for i,j\in I, i<j and {w}_{i}<w for all i\in I and w={sup}_{i}{w}_{i}. Moreover, {k}_{w}={\bigcup}_{i}{k}_{{v}_{i}} and {G}_{w}={\bigcup}_{i}{G}_{{v}_{i}} are satisfied. □