<?xml version="1.0"?>
<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a<sub>1</sub></font>) <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a<sub>1</sub></font>) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  ex <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a<sub>1</sub></font>) &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Olive">m</font></span>)</span> ) ) );<br/>
<a NAME="E1:13_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
;<br/>
<a NAME="E2:13_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:13_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1_1"><b>suppose </b></a><a NAME="E1:13_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:13_1_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1_2"><b>suppose </b></a><a NAME="E1:13_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:13_1_1_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</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">m</font> = <font color="Maroon">m</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="E5:13_1_1_1_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span>
 ;<br/><b>hence </b><a NAME="E6:13_1_1_1_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:13_1"/>
 ex <font color="Olive">F</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 
<b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<b>then consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:13_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() or (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  ex <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">x</font>) &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Olive">m</font></span>)</span> ) ) )
 ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E6:13_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) iff <font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">x</font>)</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:13_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1_2_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>)
 ;<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="E3:13_1_2_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>) &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:13_1_2_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1_2_1_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> )
 </span><b>by </b><i>, <a class="ref" href="qc_lang4.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:13_1_2_1_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:13_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_1_2_1_1_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:13_1_2_1_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:13_1_2_1_1_1_2"/><b>then </b>
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E3:13_1_2_1_1_1_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E5:13_1_2_1"/><b>then </b>
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E6:13_1_2_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>)</span>)</span>
 <b>by </b><i><a class="ref" href="qc_lang4.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:13_1_2"/>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) implies <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>)</span>)</span> )
 ;<br/>

<b>assume </b><a NAME="E4:13_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>)</span>)</span>
 ;<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="E6:13_1_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>) &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/>
<a NAME="E7:13_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1_2_2_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:13_1_2_2_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:13_1_2_2_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_1_2_2_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:13_1_2_2_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:13_1_2_2_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E3:13_1_2_2_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:13_1_2_2_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>) )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:13_1_2_2_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="qc_lang4.html#T4" target="_self">Th4</a>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:13_1_2_2_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E7:13_1_2_2_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E8:13_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>)
 <b>by </b><i><a class="ref" href="qc_lang4.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:13_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">T</font> being <a href="trees_9.html#V4">finite-branching</a> <a href="trees_2.html#NM4">DecoratedTree</a><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">t</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Olive">T</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">x</font>)</span>)</span> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_3"><b>proof </b></a><div class="add">

<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="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">t</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>assume </b><a NAME="E1:13_1_3"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>thus </b><a NAME="E2:13_1_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>)</span>)</span> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) </span>}</span> 
 ;<br/>

<b>then 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="E3:13_1_3_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) )
 ;<br/>
<a NAME="E4:13_1_3_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>)</span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="qc_lang4.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E5:13_1_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>)</span>)</span> </span>}</span> 
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:13_1_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>)</span>)</span> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_3_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>)</span>)</span> </span>}</span> 
 ;<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="E3:13_1_3_2"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>)</span>)</span> )
 ;<br/>
<a NAME="E4:13_1_3_2"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) )
 
<b>by </b><i>, , <a class="ref" href="qc_lang4.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E5:13_1_3_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">n</font></span><a href="tarski.html#K4">]</a></span> &amp; ( ( <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">x</font>) <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( not <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) ) );<br/>
<a NAME="E8:13_1"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>  ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">c</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<a NAME="E1:13_1_4"/>
( <font color="Maroon">c</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">c</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">n</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="E3:13_1_4"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="mcart_1.html#K2">`2</a>  )
 ;<br/>
<a NAME="E4:13_1_4"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">n</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="qc_lang4.html#T4" target="_self">Th4</a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1_4_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) or  not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_4_1_1"><b>suppose </b></a><a NAME="E1:13_1_4_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>)
 ;<br/><div class="add"><a NAME="E2:13_1_4_1_1"/><b>then </b>
<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#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>)</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:13_1_4_1_1"/><b>then </b>
<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> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>)
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:13_1_4_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>) <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>)
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:13_1_4_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E7:13_1_4_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>) <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E8:13_1_4_1_1"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c</font>,<font color="Olive">y</font>]
 <b>by </b><i>, , <a class="txt" href="qc_lang4.html#E1:13_1"><i><font color="Green">E22</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_4_1_2"><b>suppose </b></a><a NAME="E1:13_1_4_1_2"/>
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>)
 ;<br/><div class="add"><b>hence </b><a NAME="E2:13_1_4_1_2"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c</font>,<font color="Olive">y</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E9:13_1"/>
 ex <font color="Olive">S</font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">F<sub>1</sub></font>() st <br/>for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">c</font>,<font color="Olive">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>then consider </b><font color="Maroon">S</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E11:13_1"/><i><font color="Green">E85</font></i>: 
for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">c</font>,<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font>]
 ;<br/>
<a NAME="E12:13_1"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) holds <br/><font color="Maroon">S</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">x</font>) <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:13_1_5"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) )
 ;<br/>

<a NAME="E2:13_1_5"/><i><font color="Green">E88</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">n</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E3:13_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">n</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E4:13_1_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">n</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T11">MCART_1:11</a></i>;<br/>
<b>then consider </b><font color="Maroon">c</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E6:13_1_5"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">n</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>
<b>consider </b><font color="Maroon">x'</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">n'</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="E8:13_1_5"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">n'</font></span><a href="tarski.html#K4">]</a></span> &amp; ( ( <font color="Maroon">n'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x'</font>) &amp; <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x'</font>) <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( not <font color="Maroon">n'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x'</font>) &amp; <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) ) )
 <b>by </b><i><a class="ref" href="qc_lang4.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E9:13_1_5"/>
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">n'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:13_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="qc_lang4.html#T6" target="_self">Th6</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E10:13_1_5"/>
<font color="Maroon">S</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>) <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="qc_lang4.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:13_1"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">x</font>) &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/>( <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) iff ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:13_1_6"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>) &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 ;<br/>

<b>thus </b><a NAME="E2:13_1_6"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) implies ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1_6_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>)
 ;<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="E3:13_1_6_1"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E4:13_1_6_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="qc_lang4.html#T6" target="_self">Th6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1_6_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> )
 </span><b>by </b><i><a class="txt" href="qc_lang4.html#E1:13_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="qc_lang4.html#T6" target="_self">Th6</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_6_1_1_1"><b>suppose </b></a><a NAME="E1:13_1_6_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:13_1_6_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_1_6_1_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:13_1_6_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:13_1_6_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E3:13_1_6_1_1_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E3:13_1_6"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> implies <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_6_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1_6_2"/><i><font color="Green">E96</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 ;<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="E3:13_1_6_2"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">x</font>) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E4:13_1_6_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="qc_lang4.html#T6" target="_self">Th6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_6_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1_6_2_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_6_2_1_1"><b>suppose </b></a><a NAME="E1:13_1_6_2_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:13_1_6_2_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_1_6_2_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>)
 <b>by </b><i><a class="ref" href="qc_lang4.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_6_2_1_2"><b>suppose </b></a><a NAME="E1:13_1_6_2_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:13_1_6_2_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E3:13_1_6_2_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:13_1_6_2_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E1:13_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E5:13_1_6_2_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>)
 <b>by </b><i><a class="ref" href="qc_lang4.html#T6" target="_self">Th6</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E14:13_1"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">k1</font>, <font color="Olive">k2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k2</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">d</font>) holds <br/><font color="Olive">k1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">d</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">k1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">k2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:13_1_7"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font> &amp; <font color="Maroon">k2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">d</font>) )
 ;<br/>

<a NAME="E2:13_1_7"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">d</font>) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  ex <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">d</font>) &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Olive">m</font></span>)</span> ) )
 
<b>by </b><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="E4:13_1_7"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">d</font>)
 ;<br/>
<a NAME="E5:13_1_7"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font> &amp; <font color="Maroon">k2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 
<b>by </b><i>, <a class="txt" href="qc_lang4.html#E1:13_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="qc_lang4.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E6:13_1_7"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:13_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<b>hence </b><a NAME="E7:13_1_7"/>
<font color="Maroon">k1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">d</font>)
 <b>by </b><i>, <a class="ref" href="qc_lang4.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">T</font> being   <a href="trees_2.html#NM6">DecoratedTree</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E16:13_1"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive">t</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> holds <br/> (  <a href="trees_2.html#K1">succ</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Olive">t</font>) </span>}</span>  &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Olive">t</font>) holds <br/><font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Olive">t</font></span>)</span>,<font color="Olive">n</font> ) ) ) )
 <b>from </b><i><a class="ref" href="trees_2.html#S8">TREES_2:sch 8</a>();<br/></i>
<a NAME="E17:13_1"/>
<font color="Maroon">T</font> is <a href="trees_9.html#V4">finite-branching</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_8"><b>proof </b></a><div class="add">

<a NAME="E1:13_1_8"/>
for <font color="Olive">t</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> holds   <a href="trees_2.html#K1">succ</a> <font color="Olive">t</font> is <a href="finset_1.html#V1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</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/>

<a NAME="E1:13_1_8_1"/><i><font color="Green">E103</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font>) </span>}</span> 
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:13_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span> );<br/>
<a NAME="E2:13_1_8_1"/><i><font color="Green">E104</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>)</span>)</span> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
;<br/>
<a NAME="E3:13_1_8_1"/><i><font color="Green">E105</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>)</span>)</span> holds <br/> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_8_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:13_1_8_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>)</span>)</span>
 ;<br/>

<a NAME="E2:13_1_8_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:13_1_8_1_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</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">m</font> = <font color="Maroon">m</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/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:13_1_8_1_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/>
<b>thus </b><a NAME="E8:13_1_8_1_1"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">k</font>,<font color="Olive">x</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:13_1_8_1"/>
 ex <font color="Olive">L</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>)</span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>)</span>)</span> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>,<font color="Olive">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>] ) )
 
<b>from </b><i><a class="ref" href="finseq_1.html#S1">FINSEQ_1:sch 1</a>(, );<br/></i>
<b>then consider </b><font color="Maroon">L</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E6:13_1_8_1"/><i><font color="Green">E109</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>)</span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>)</span>)</span> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>,<font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>] ) )
 ;<br/>
<a NAME="E7:13_1_8_1"/><i><font color="Green">E110</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>) holds <br/><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_8_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:13_1_8_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>) )
 ;<br/>

<a NAME="E2:13_1_8_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>)</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</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="E4:13_1_8_1_2"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E5:13_1_8_1_2"/>
<font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E8:13_1_8_1"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_8_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:13_1_8_1_3"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_8_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_8_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font>
 ;<br/>

<b>then 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="E3:13_1_8_1_3_1"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font>) )
 <b>by </b><i><a class="ref" href="qc_lang4.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E4:13_1_8_1_3_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>)</span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:13_1_8_1_3_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>) )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:13_1_8_1_3_1"/><b>then </b>
<font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:13_1_8_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font>
 <b>by </b><i>, , <a class="txt" href="qc_lang4.html#E2:13_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:13_1_8_1_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_8_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_8_1_3_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon">m</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:13_1_8_1_3_2"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E5:13_1_8_1_3_2"/><i><font color="Green">E115</font></i>: 
( 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>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>) )
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:13_1_8_1_3_2"/>
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:13_1_8_1_3_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">k</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">k</font> = <font color="Maroon">k</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="E10:13_1_8_1_3_2"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font>) )
 
<b>by </b><i>, , , , <a class="txt" href="qc_lang4.html#E2:13_1"><i><font color="Green">E29</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E11:13_1_8_1_3_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="qc_lang4.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:13_1_8_1"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:13_1_8"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> is <a href="trees_9.html#V2">finite-branching</a>
 
<b>by </b><i><a class="ref" href="trees_9.html#D2">TREES_9:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:13_1_8"/>
<font color="Maroon">T</font> is <a href="trees_9.html#V4">finite-branching</a>
 <b>by </b><i><a class="ref" href="trees_9.html#D4">TREES_9:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">T</font> = <font color="Maroon">T</font> as  <a href="trees_9.html#V4">finite-branching</a> <a href="trees_2.html#NM6">DecoratedTree</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</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">w</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>assume </b><a NAME="E1:13_1_9"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font>
 ;<br/><a NAME="E2:13_1_9"/><i><font color="Green">E118</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">t</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_9_1"><b>proof </b></a><div class="add">

<a NAME="E1:13_1_9_1"/><i><font color="Green">E119</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</font>) <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</font>)</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:13_1_9_1"/><i><font color="Green">E120</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="trees_9.html#K1">succ</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="trees_9.html#K1">succ</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:13_1_9_1"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">w</font>) </span>}</span> 
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:13_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="qc_lang4.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E4:13_1_9_1"/><b>then </b><i><font color="Green">E121</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</font>)</span>)</span> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:13_1_9_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</font>) <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="trees_9.html#K1">succ</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_9_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:13_1_9_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</font>) <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="trees_9.html#K1">succ</a> </span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_9_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n'</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_9_1_1_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">n'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</font>)
 ;<br/>

<a NAME="E2:13_1_9_1_1_1"/><i><font color="Green">E123</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</font>)</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</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">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</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="E4:13_1_9_1_1_1"/><i><font color="Green">E124</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>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</font>) )
 <b>by </b><i>, </i>;<br/>
<a NAME="E5:13_1_9_1_1_1"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font>
 
<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="E7:13_1_9_1_1_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <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="E9:13_1_9_1_1_1"/>
<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#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</font>)</span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E2:13_1"><i><font color="Green">E29</font></i></a>, , </i>;<br/>
<a NAME="E10:13_1_9_1_1_1"/><b>then </b>
<font color="Maroon">t</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="trees_2.html#K1">succ</a> <font color="Maroon">t</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:13_1_9_1_1_1"/>
<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"><font color="Maroon">t</font> <a href="trees_9.html#K1">succ</a> </span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:13_1_9_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="trees_9.html#K1">succ</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_9_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n'</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_9_1_1_2"/><i><font color="Green">E126</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"><font color="Maroon">t</font> <a href="trees_9.html#K1">succ</a> </span>)</span>
 ;<br/>

<a NAME="E2:13_1_9_1_1_2"/><i><font color="Green">E127</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="trees_9.html#K1">succ</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</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">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="trees_9.html#K1">succ</a> </span>)</span> ) </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="E4:13_1_9_1_1_2"/><i><font color="Green">E128</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>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="trees_9.html#K1">succ</a> </span>)</span> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E5:13_1_9_1_1_2"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font>
 
<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="E7:13_1_9_1_1_2"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <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="E9:13_1_9_1_1_2"/>
<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">t</font> <a href="trees_9.html#K1">succ</a> </span>)</span>
 
<b>by </b><i>, <a class="txt" href="qc_lang4.html#E2:13_1"><i><font color="Green">E29</font></i></a>, , </i>;<br/>
<a NAME="E10:13_1_9_1_1_2"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">t</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/>
<a NAME="E11:13_1_9_1_1_2"/>
<font color="Maroon">t</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="trees_2.html#K1">succ</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E5:13_1"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>then 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="E13:13_1_9_1_1_2"/><i><font color="Green">E131</font></i>: 
( <font color="Maroon">t</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">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</font>)</span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E14:13_1_9_1_1_2"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</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>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/>
<b>hence </b><a NAME="E15:13_1_9_1_1_2"/>
<font color="Maroon">n'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</font>)
 <b>by </b><i>, , , , <a class="ref" href="trees_1.html#T23">TREES_1:23</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:13_1_9_1"/>
 <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">t</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</font>)
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E3:13_1_9"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</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">t</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="trees_9.html#K2">succ</a> <font color="Maroon">T</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</font>) <a href="funct_1.html#K1">.</a> <font color="Olive">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_9_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:13_1_9_2"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">m</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">t</font></span>)</span>
 ;<br/>

<a NAME="E2:13_1_9_2"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</font>)</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:13_1_9_2"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</font>)</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</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">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</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">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="E5:13_1_9_2"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</font>) )
 <b>by </b><i/>;<br/>
<a NAME="E6:13_1_9_2"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font>
 
<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="E8:13_1_9_2"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <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="E10:13_1_9_2"/>
<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">t</font> <a href="trees_9.html#K1">succ</a> </span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:13_1_9_2"/><b>then </b><i><font color="Green">E136</font></i>: 
<font color="Maroon">t</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>consider </b><font color="Maroon">l</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="E13:13_1_9_2"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>) &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="qc_lang4.html#T6" target="_self">Th6</a>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/>
<a NAME="E14:13_1_9_2"/>
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="qc_lang4.html#T6" target="_self">Th6</a>, , , </i>;<br/>
<a NAME="E15:13_1_9_2"/><b>then </b><i><font color="Green">E138</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E16:13_1_9_2"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_9_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_9_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1_9_2_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_9_2_1_1_1"><b>suppose </b></a><a NAME="E1:13_1_9_2_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:13_1_9_2_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_1_9_2_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_9_2_1_1_2"><b>suppose </b></a><a NAME="E1:13_1_9_2_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:13_1_9_2_1_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E3:13_1_9_2_1_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:13_1_9_2_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E5:13_1_9_2_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<span class="p1">(<span class="default"><a href="trees_9.html#K2">succ</a> <font color="Maroon">T</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><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></span>)</span>
<b>by </b><i><a class="txt" href="qc_lang4.html#E2:13_1"><i><font color="Green">E29</font></i></a>, </i>
<br/>.= 
<font color="Maroon">S</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:13_1"><i><font color="Green">E22</font></i></a>, , <a class="txt" href="qc_lang4.html#E5:13_1"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</font>) <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="qc_lang4.html#T6" target="_self">Th6</a>, , <a class="txt" href="qc_lang4.html#E5:13_1"><i><font color="Green">E32</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E18:13_1_9_2"/>
<span class="p1">(<span class="default"><a href="trees_9.html#K2">succ</a> <font color="Maroon">T</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</font>) <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:13_1_9"/>
 <a href="trees_9.html#K2">succ</a> <font color="Maroon">T</font>,<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">w</font>)
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E20:13_1"/>
 ex <font color="Olive">T</font> being <a href="trees_9.html#V4">finite-branching</a> <a href="trees_2.html#NM6">DecoratedTree</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive">T</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive">t</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Olive">T</font><br/> for <font color="Olive">w</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">T</font> <a href="trees_2.html#K3">.</a> <font color="Olive">t</font> holds <br/> <a href="trees_9.html#K2">succ</a> <font color="Olive">T</font>,<font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">w</font>) ) )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E1:13_1"><i><font color="Green">E22</font></i></a></i>;<br/>


</div>
