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

<b>let </b><font color="Maroon">e1</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">e2</font> be  <a href="finset_1.html#V1">finite</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">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p</font> be  <a href="trees_3.html#V6">DTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:35"/><i><font color="Green">E29</font></i>: 
<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#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">e1</font>
 <b>and </b><br/><a NAME="E2:35"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font>
 ;<br/>

<b>set </b><font color="Maroon">o</font> = <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>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">e2</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E4:35"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E5:35"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</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">p</font></span>)</span> holds <br/><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E6:35"/><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</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">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:35"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </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="E1:35_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font>
 ;<br/><b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:35_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">e2</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/><a NAME="E4:35_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><a NAME="E5:35_1"/><b>then </b>
( <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> or <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>hence </b><a NAME="E6:35_1"/>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is   <a href="trees_2.html#NM4">DecoratedTree</a>
 <b>by </b><i>, , <a class="ref" href="trees_3.html#T26">TREES_3:26</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as  <a href="trees_3.html#V6">DTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="trees_3.html#T26">TREES_3:26</a></i>;<br/>
<b>take </b>
<font color="Maroon">q</font>
;<br/>

<a NAME="E10:35"/>
( <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>  <a href="xboole_0.html#K1">{}</a>  or  ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">T</font> being  <a href="trees_2.html#NM4">DecoratedTree</a> ex <font color="Olive">q</font> being  <a href="trees_4.html#NM1">Node</a> of <font color="Olive">T</font> st <br/>( <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; <font color="Olive">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <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="Olive">i</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Olive">q</font> ) )
 
<b>by </b><i>, , <a class="ref" href="trees_4.html#T11">TREES_4:11</a></i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">T</font> being   <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">w</font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E12:35"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E13:35"/>
<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E14:35"/><i><font color="Green">E52</font></i>: 
<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">j</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w</font>
 ;<br/>
<a NAME="E15:35"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">j</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">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="trees_1.html#T6">TREES_1:6</a></i>;<br/>
<b>then </b><i><font color="Green">E53</font></i>: <font color="Maroon">j</font> = 
<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="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>
<br/>.= 
<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>
;<br/>
<a NAME="E17:35"/><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">p</font> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E18:35"/><b>then </b><i><font color="Green">E55</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#K4">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E19:35"/><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">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E58</font></i>: <font color="Maroon">q</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="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">e2</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">e2</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/>
<b>consider </b><font color="Maroon">qq</font> being  <a href="trees_3.html#V6">DTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E22:35"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qq</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="trees_3.html#K13">tree</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">qq</font></span>)</span> )
 <b>by </b><i><a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">pp</font> being  <a href="trees_3.html#V6">DTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E24:35"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pp</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="trees_3.html#K13">tree</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">pp</font></span>)</span> )
 <b>by </b><i><a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">dqq</font> =  <a href="funct_6.html#K2">doms</a> <font color="Maroon">qq</font> as  <a href="trees_3.html#V4">Tree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> ;<br/>
<i><font color="Green">E86</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">pp</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="trees_3.html#T40">TREES_3:40</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">qq</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="trees_3.html#T40">TREES_3:40</a></i>
;<br/>
<a NAME="E27:35"/><i><font color="Green">E87</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">pp</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="trees_3.html#T39">TREES_3:39</a></i>;<br/>
<div><i><font color="Green">E104</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:35_5"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">pp</font></span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><b>then </b><i><font color="Green">E106</font></i>: <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">e2</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><b>reconsider </b><font color="Maroon">pii</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="trees_2.html#NM4">DecoratedTree</a> <b>by </b><i>, , <a class="ref" href="trees_3.html#T26">TREES_3:26</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">pp</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">pii</font>
<b>by </b><i>, , , <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">qq</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , , , , <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E29:35"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">qq</font></span>)</span> <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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">e2</font>
 
<b>by </b><i>, , , , <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/>
<a NAME="E30:35"/><b>then </b><i><font color="Green">E107</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">e1</font></span>)</span> <a href="trees_1.html#K5">with-replacement</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>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">e2</font></span>)</span>
 
<b>by </b><i>, , , , , , , </i>;<br/>
<a NAME="E31:35"/>
for <font color="Olive">f</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   holds <br/>(  not <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">e1</font></span>)</span> <a href="trees_1.html#K5">with-replacement</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>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">e2</font></span>)</span> or ( not <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="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">f</font> ) or  ex <font color="Olive">r</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">e2</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</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="finseq_1.html#K8">^</a> <font color="Olive">r</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:35_6"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">e1</font></span>)</span> <a href="trees_1.html#K5">with-replacement</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>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">e2</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">o'</font> = <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> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">e1</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_6_1"/>
(  not <font color="Maroon">o'</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">f</font> or  ex <font color="Olive">t1</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">e2</font> st <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">t1</font> )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_6_1_1"><b>suppose </b></a><a NAME="E1:35_6_1_1"/><i><font color="Green">E109</font></i>: 
not <font color="Maroon">o'</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:35_6_1_1"/><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_6_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_6_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_6_1_1_1_1"/>
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  ex <font color="Olive">w</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">u</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">dqq</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">dqq</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">w</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Olive">u</font> ) )
 </span><b>by </b><i>, , , <a class="ref" href="trees_3.html#D15">TREES_3:def 15</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_6_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:35_6_1_1_1_1_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> = 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>
<br/>.= 
<font color="Maroon">e1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font>
<b>by </b><i>, , <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_6_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:35_6_1_1_1_1_2"/>
 ex <font color="Olive">w</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">u</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">dqq</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">dqq</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">w</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Olive">u</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">w</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">u</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:35_6_1_1_1_1_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">q</font></span>)</span>
 <b>and </b><br/><a NAME="E4:35_6_1_1_1_1_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E5:35_6_1_1_1_1_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">w</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">u</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="finseq_1.html#T50">FINSEQ_1:50</a></i>;<br/><b>reconsider </b><font color="Maroon">ww</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">w</font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E8:35_6_1_1_1_1_2"/>
<font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="trees_1.html#T8">TREES_1:8</a></i>;<br/><a NAME="E9:35_6_1_1_1_1_2"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T2">XCMPLX_1:2</a></i>;<br/><a NAME="E10:35_6_1_1_1_1_2"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="trees_3.html#T40">TREES_3:40</a></i>;<br/><a NAME="E11:35_6_1_1_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">w</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">p</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:35_6_1_1_1_1_2"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">w</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">p</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E13:35_6_1_1_1_1_2"/><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q</font></span>)</span> <a href="trees_2.html#K5">|</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">w</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="pre_circ.html#E4:35"><i><font color="Green">E34</font></i></a>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/><a NAME="E14:35_6_1_1_1_1_2"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">w</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"><a href="funct_6.html#K2">doms</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="trees_3.html#T39">TREES_3:39</a></i>;<br/><a NAME="E15:35_6_1_1_1_1_2"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">w</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"><a href="funct_6.html#K2">doms</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="trees_3.html#T39">TREES_3:39</a></i>;<br/><i><font color="Green">E131</font></i>: <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">e2</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><b>reconsider </b><font color="Maroon">pw1</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="trees_2.html#NM4">DecoratedTree</a> <b>by </b><i>, <a class="ref" href="trees_3.html#T26">TREES_3:26</a></i>;<br/><a NAME="E18:35_6_1_1_1_1_2"/><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="trees_1.html#K4">|</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">w</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , </i>;<br/><b>consider </b><font color="Maroon">pp</font> being  <a href="trees_3.html#V6">DTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E20:35_6_1_1_1_1_2"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pp</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="trees_3.html#K13">tree</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">pp</font></span>)</span> )
 <b>by </b><i><a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/><i><font color="Green">E135</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="trees_1.html#K4">|</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">w</font></span><a href="finseq_1.html#K12">*&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="pre_circ.html#T3" target="_self">Th3</a>, <a class="txt" href="pre_circ.html#E6:35"><i><font color="Green">E41</font></i></a>, </i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">pw1</font>
<b>by </b><i>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , , <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q</font></span>)</span> <a href="trees_2.html#K5">|</a> <font color="Maroon">ww</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
<b>by </b><i><a class="txt" href="pre_circ.html#E2:35"><i><font color="Green">E31</font></i></a>, , <a class="txt" href="pre_circ.html#E5:35"><i><font color="Green">E40</font></i></a>, <a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font></span>)</span> <a href="trees_2.html#K5">|</a> <font color="Maroon">ww</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
<b>by </b><i>, <a class="txt" href="pre_circ.html#E4:35"><i><font color="Green">E34</font></i></a>, , , <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>
<br/>.= 
<font color="Maroon">e1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font>
<b>by </b><i>, <a class="txt" href="pre_circ.html#E2:35"><i><font color="Green">E31</font></i></a>, , <a class="txt" href="pre_circ.html#E7:35"><i><font color="Green">E42</font></i></a>, <a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>assume </b><a NAME="E3:35_6_1_1"/>
( <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="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">f</font> or <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> )
 ;<br/><b>hence </b><a NAME="E4:35_6_1_1"/>
 ex <font color="Olive">r</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">e2</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</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="finseq_1.html#K8">^</a> <font color="Olive">r</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> )
 <b>by </b><i>, <a class="txt" href="pre_circ.html#E1:35"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_6_1_2"><b>suppose </b></a><a NAME="E1:35_6_1_2"/>
 ex <font color="Olive">t1</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">e2</font> st <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">t1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">r</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">e2</font><b> such that </b><br/><a NAME="E3:35_6_1_2"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</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="finseq_1.html#K8">^</a> <font color="Maroon">r</font>
 ;<br/><b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>assume </b><a NAME="E5:35_6_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="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">f</font> or <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> )
 ;<br/><b>take </b>
<font color="Maroon">r</font>
;<br/><b>thus </b><a NAME="E6:35_6_1_2"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">e2</font>
 ;<br/><a NAME="E7:35_6_1_2"/><i><font color="Green">E138</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q</font></span>)</span> <a href="trees_2.html#K5">|</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">q</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>
 <b>by </b><i>, , , <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/><a NAME="E8:35_6_1_2"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="trees_1.html#K4">|</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>, , <a class="txt" href="pre_circ.html#E1:35"><i><font color="Green">E29</font></i></a>, <a class="ref" href="trees_1.html#T66">TREES_1:66</a></i>;<br/><b>thus </b><a NAME="E9:35_6_1_2"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</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="finseq_1.html#K8">^</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E10:35_6_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font>
 <b>by </b><i>, , , <a class="txt" href="pre_circ.html#E2:35"><i><font color="Green">E31</font></i></a>, <a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E32:35"/>
<font color="Maroon">e1</font> <a href="trees_2.html#K8">with-replacement</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>,<font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">q</font>
 <b>by </b><i>, , <a class="ref" href="trees_2.html#D12">TREES_2:def 12</a></i>;<br/>

<b>thus </b><a NAME="E33:35"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E34:35"/>
<font color="Maroon">q</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">e2</font>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E35:35"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E36:35"/><b>then </b>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">e2</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E37:35"/>
(  not <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/>


</div>
