<?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> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span> &amp; <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> );<br/>
<b>consider </b><font color="Maroon">W</font> being   <a href="trees_1.html#NM1">Tree</a><b> such that </b><br/><a NAME="E2:43_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a> holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> iff ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> or  ex <font color="Olive">q</font> being  <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T</font> ex <font color="Olive">r</font> being  <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T'</font> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>] &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r</font> ) ) )
 <b>from </b><i><a class="ref" href="trees_4.html#S1" target="_self">TREES_4:sch 1</a>();<br/></i>
<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>( ( <font color="Maroon">a<sub>1</sub></font> is   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T</font> &amp; (  not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span> or <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> ) &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) or  ex <font color="Olive">p</font> being  <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T</font> ex <font color="Olive">q</font> being  <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T'</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">q</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span> &amp; <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">T'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> ) );<br/>
<a NAME="E3:43_1_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">z</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:43_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">w</font> = <font color="Maroon">z</font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">W</font> ;<br/>
<div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">q</font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T</font>, <font color="Maroon">r</font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T'</font><b> such that </b><a NAME="E1:43_1_1_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span> &amp; <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font> )
 ;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">T'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font>;<br/><b>take </b><font color="Maroon">q</font> = <font color="Maroon">q</font>;<br/><b>take </b><font color="Maroon">r</font> = <font color="Maroon">r</font>;<br/><b>thus </b><a NAME="E2:43_1_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span> &amp; <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_1_1_2"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">q</font> being  <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T</font><br/> for <font color="Olive">r</font> being  <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T'</font>  holds <br/>(  not <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span> or  not <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or  not <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r</font> )
 ;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>;<br/><b>thus </b><a NAME="E2:43_1_1_1_2"/>
<font color="Maroon">z</font> is   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T</font>
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">w</font> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T</font> <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">e</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T'</font> <b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/><a NAME="E5:43_1_1_1_2"/>
<font color="Maroon">w</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><b>hence </b><a NAME="E6:43_1_1_1_2"/>
( (  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span> or <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> ) &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:43_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">z</font>,<font color="Olive">y</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:43_1_1"/><i><font color="Green">E51</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">z</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font>] ) )
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="trees_2.html#NM4">DecoratedTree</a> <b>by </b><i>, <a class="ref" href="trees_2.html#D8">TREES_2:def 8</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E7:43_1_1"/>
for <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a> holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> iff ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> or  ex <font color="Olive">q</font> being  <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T</font> ex <font color="Olive">r</font> being  <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T'</font> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span> &amp; <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r</font> ) ) )
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E8:43_1_1"/>
for <font color="Olive">p</font> being  <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T</font>  st (  not <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span> or <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> ) holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:43_1_1_2"/><i><font color="Green">E53</font></i>: 
(  not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span> or <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 ;<br/>

<a NAME="E2:43_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_2_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">p'</font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T</font>, <font color="Maroon">q</font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T'</font><b> such that </b><a NAME="E1:43_1_1_2_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> &amp; <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span> &amp; <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> )
 ;<br/><a NAME="E2:43_1_1_2_1"/>
( <font color="Maroon">p'</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">p</font> &amp; not <font color="Maroon">p'</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="trees_1.html#T8">TREES_1:8</a>, <a class="ref" href="trees_1.html#D8">TREES_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:43_1_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:43_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>
 <b>by </b><i>, </i>;<br/>


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

<b>let </b><font color="Maroon">p</font> be   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T'</font>;<br/>



<b>assume </b><a NAME="E9:43_1_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span> &amp; <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 ;<br/>

<a NAME="E10:43_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_1_3"/>
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> is   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T</font>
 ;<br/><a NAME="E2:43_1_1_3"/><b>then </b>
( not <font color="Maroon">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> &amp; <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> )
 <b>by </b><i>, <a class="ref" href="trees_1.html#T8">TREES_1:8</a>, <a class="ref" href="trees_1.html#D8">TREES_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:43_1_1_3"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">p'</font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T</font>, <font color="Maroon">q'</font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">T'</font><b> such that </b><br/><a NAME="E13:43_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q'</font> &amp; <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span> &amp; <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">T'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q'</font> )
 <b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">p'</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">q'</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">T</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/><b>assume </b><a NAME="E1:43_1_1_4"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q'</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">T</font> &amp; <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">T</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p'</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_1_4_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p'</font>
 ;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:43_1_1_4_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T64">FINSEQ_1:64</a></i>;<br/><a NAME="E4:43_1_1_4_1"/>
<font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">p'</font>
 <b>by </b><i>, <a class="ref" href="trees_1.html#T8">TREES_1:8</a></i>;<br/><a NAME="E5:43_1_1_4_1"/><b>then </b>
<font color="Maroon">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">p'</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E6:43_1_1_4_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="trees_1.html#D8">TREES_1:def 8</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">r</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E4:43_1_1_4"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">p'</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T64">FINSEQ_1:64</a></i>;<br/><a NAME="E5:43_1_1_4"/>
<font color="Maroon">p'</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="trees_1.html#T8">TREES_1:8</a></i>;<br/><a NAME="E6:43_1_1_4"/><b>then </b>
<font color="Maroon">p'</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E7:43_1_1_4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="trees_1.html#D8">TREES_1:def 8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:43_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E16:43_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">T'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/>


</div>
