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<b>let </b><font color="Maroon">T</font> be  <a href="trees_9.html#V4">finite-branching</a> <a href="trees_2.html#NM4">DecoratedTree</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">y'</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:12"/>
<font color="Maroon">y'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2">succ</a> <font color="Maroon">T</font>,<font color="Maroon">x</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">n'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">n'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2">succ</a> <font color="Maroon">T</font>,<font color="Maroon">x</font></span>)</span> &amp; <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2">succ</a> <font color="Maroon">T</font>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n'</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:12"/><i><font color="Green">E29</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2">succ</a> <font color="Maroon">T</font>,<font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E6:12"/>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">m</font> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">m'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:12"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">n'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m'</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m'</font> &amp; <font color="Maroon">m'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E9:12"/>
<font color="Maroon">m'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:12"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m'</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E13:12"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_9.html#K1">succ</a> </span>)</span>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E14:12"/><b>then </b>
<font color="Maroon">x</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">x'</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font><b> such that </b><br/><a NAME="E16:12"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/>
<a NAME="E17:12"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E18:12"/>
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font>
 
<b>by </b><i>, , , , </i>;<br/>
<b>hence </b><a NAME="E19:12"/>
 ex <font color="Olive">x'</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> st <br/>( <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x'</font> &amp; <font color="Olive">x'</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">x</font> )
 <b>by </b><i/>;<br/>


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