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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="trees_3.html#V4">Tree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:33"/><i><font color="Green">E10</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:33"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:33"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<a NAME="E4:33"/>
<font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="pre_circ.html#E2:33"><i><font color="Green">E11</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:33"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="trees_1.html#NM1">Tree</a>;<br/>

<b>assume </b><a NAME="E6:33"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E7:33"/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is   <a href="trees_1.html#NM1">Tree</a>
 
<b>by </b><i><a class="txt" href="pre_circ.html#E2:33"><i><font color="Green">E11</font></i></a>, <a class="ref" href="trees_3.html#T24">TREES_3:24</a></i>;<br/>
<a NAME="E8:33"/><b>then </b><i><font color="Green">E15</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<a NAME="E9:33"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>
<a NAME="E10:33"/><b>then </b><i><font color="Green">E16</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K13">tree</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="pre_circ.html#E5:33"><i><font color="Green">E13</font></i></a>, <a class="txt" href="pre_circ.html#E8:33"><i><font color="Green">E15</font></i></a>, <a class="ref" href="trees_3.html#D15">TREES_3:def 15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K13">tree</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Olive">b<sub>2</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="pre_circ.html#E1:33_1_1"><i><font color="Green">E17</font></i></a>, <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="33_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1"/>
( ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K13">tree</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>6</sub></font> ) or ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a>, <a class="ref" href="xboole_1.html#T115">XBOOLE_1:115</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Olive">b<sub>2</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">c<sub>8</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:33_1_1_1_2"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_2_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:33_1_1_1_2_1_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="pre_circ.html#E3:33_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="finseq_1.html#T50">FINSEQ_1:50</a></i>;<br/><b>take </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font>;<br/><b>thus </b><a NAME="E3:33_1_1_1_2_1_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="pre_circ.html#E6:33"><i><font color="Green">E14</font></i></a>, <a class="txt" href="pre_circ.html#E3:33_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="pre_circ.html#E1:33_1_1_1_2_1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:33_1_1_1_2_1_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:33_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></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="E3:33_1_1_1_2_1_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="pre_circ.html#E1:33"><i><font color="Green">E10</font></i></a>, <a class="txt" href="pre_circ.html#E3:33_1_1_1_2"><i><font color="Green">E18</font></i></a>, <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="E4:33_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:33_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="pre_circ.html#E3:33"><i><font color="Green">E12</font></i></a>, <a class="txt" href="pre_circ.html#E2:33_1_1_1_2_1_1_2"><i><font color="Green">E20</font></i></a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K13">tree</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="pre_circ.html#E1:33"><i><font color="Green">E10</font></i></a>, <a class="txt" href="pre_circ.html#E3:33_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="trees_3.html#D15">TREES_3:def 15</a></i>;<br/><b>assume </b><a NAME="E7:33_1_1_1_2_1_1_2"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E8:33_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E10:33_1_1_1_2_1_1_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="trees_1.html#T8">TREES_1:8</a></i>;<br/><font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="pre_circ.html#E10:33_1_1_1_2_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="pre_circ.html#E3:33_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>
;<br/><b>hence </b><a NAME="E12:33_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pre_circ.html#E1:33_1_1_1_2_1_1_2"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:33_1_1_1_2"/>
( ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K13">tree</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>6</sub></font> ) or ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>assume </b><a NAME="E2:33_1"/><i><font color="Green">E17</font></i>: 
( ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K13">tree</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>6</sub></font> ) or ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_2"/>
( ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K13">tree</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>6</sub></font> ) or ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="pre_circ.html#E2:33_1"><i><font color="Green">E17</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_2_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K13">tree</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:33_1_2_1"/><i><font color="Green">E19</font></i>: 
not <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_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="E1:33_1_2_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Olive">b<sub>2</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="pre_circ.html#E1:33_1_2_1"><i><font color="Green">E18</font></i></a>, <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="33_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K13">tree</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Olive">b<sub>2</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">c<sub>8</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:33_1_2_1_1_1_2"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_2_1_1_1_2_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_1_2_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:33_1_2_1_1_1_2_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="pre_circ.html#E1:33"><i><font color="Green">E10</font></i></a>, <a class="txt" href="pre_circ.html#E3:33_1_2_1_1_1_2"><i><font color="Green">E20</font></i></a>, <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="E3:33_1_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:33_1_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is   <a href="trees_1.html#NM1">Tree</a>
 <b>by </b><i><a class="ref" href="trees_3.html#T24">TREES_3:24</a></i>;<br/><a NAME="E5:33_1_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="pre_circ.html#E1:33_1_2_1_1_1_2_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/><b>hence </b><a NAME="E6:33_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K13">tree</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="pre_circ.html#E1:33"><i><font color="Green">E10</font></i></a>, <a class="txt" href="pre_circ.html#E3:33_1_2_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="trees_3.html#D15">TREES_3:def 15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_1_2_1_1_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_2_1_1_1_2_1_1_2_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:33_1_2_1_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="pre_circ.html#E3:33_1_2_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="trees_1.html#T8">TREES_1:8</a></i>;<br/><a NAME="E3:33_1_2_1_1_1_2_1_1_2_1"/>
not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="pre_circ.html#E3:33_1_2_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="pre_circ.html#E1:33_1_2_1_1_1_2_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="pre_circ.html#E1:33_1_2_1_1_1_2_1_1_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="trees_1.html#T5">TREES_1:5</a></i>;<br/><b>hence </b><a NAME="E4:33_1_2_1_1_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pre_circ.html#E2:33_1_2_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="pre_circ.html#E2:33_1_2_1_1_1_2_1_1_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/></div><b>end;</b></div><a NAME="E3:33_1_2_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></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="E4:33_1_2_1_1_1_2_1_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="pre_circ.html#E3:33_1_2_1_1_1_2"><i><font color="Green">E20</font></i></a>, <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="E5:33_1_2_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:33_1_2_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="pre_circ.html#E3:33"><i><font color="Green">E12</font></i></a>, <a class="txt" href="pre_circ.html#E3:33_1_2_1_1_1_2_1_1_2"><i><font color="Green">E22</font></i></a></i>;<br/><b>hence </b><a NAME="E7:33_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K13">tree</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="pre_circ.html#E1:33"><i><font color="Green">E10</font></i></a>, <a class="txt" href="pre_circ.html#E3:33_1_2_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="trees_3.html#D15">TREES_3:def 15</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:33_1_2_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K13">tree</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K13">tree</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_2"><b>suppose </b></a><a NAME="E1:33_1_2_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_2_2"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><b>thus </b><a NAME="E4:33_1_2_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K13">tree</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="pre_circ.html#E1:33"><i><font color="Green">E10</font></i></a>, <a class="txt" href="pre_circ.html#E5:33"><i><font color="Green">E13</font></i></a>, <a class="txt" href="pre_circ.html#E6:33"><i><font color="Green">E14</font></i></a>, <a class="txt" href="pre_circ.html#E3:33_1_2_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="trees_3.html#D15">TREES_3:def 15</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E12:33"/>
 <a href="trees_3.html#K13">tree</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13">tree</a> <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="pre_circ.html#E10:33"><i><font color="Green">E16</font></i></a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>


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