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

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="trees_1.html#E1:50"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E2:50_1_1"/>
not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<a NAME="E3:50_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:50_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>2</sub></font> </span>}</span>  holds <br/>( <font color="Olive">b<sub>1</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:50_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/>

<a NAME="E2:50_1_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>1</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:50_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>thus </b><a NAME="E4:50_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="trees_1.html#E2:50_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:50_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="trees_1.html#V1" target="_self">Tree-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:50_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="trees_1.html#D5" target="_self">TREES_1:def 5</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:50_1_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:50_1_1_2_1"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_1_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/><a NAME="E2:50_1_1_2_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><b>hence </b><a NAME="E3:50_1_1_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:50_1_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 <b>by </b><i><a class="txt" href="trees_1.html#E5:50_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="trees_1.html#E2:50_1_1_2_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>


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

<b>thus </b><a NAME="E2:50_1_1_2"/>
for <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 <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/> <a href="trees_1.html#K1" target="_self">ProperPrefixes</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_2"><b>proof </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/>

<b>assume </b><a NAME="E1:50_1_1_2_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_1_2_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/><a NAME="E2:50_1_1_2_2_1"/><b>then </b><i><font color="Green">E35</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><a NAME="E3:50_1_1_2_2_1"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="trees_1.html#K1" target="_self">ProperPrefixes</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="trees_1.html#D5" target="_self">Def5</a></i>;<br/><b>thus </b><a NAME="E4:50_1_1_2_2_1"/>
 <a href="trees_1.html#K1" target="_self">ProperPrefixes</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:50_1_1_2_2_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K1" target="_self">ProperPrefixes</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:50_1_1_2_2_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="trees_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="trees_1.html#E3:50_1_1_2_2_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="trees_1.html#E1:50_1_1_2_2_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="trees_1.html#E3:50_1_1_2_2_1_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E5:50_1_1_2_2_1_1"/>
not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="trees_1.html#E2:50_1_1_2_2_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="trees_1.html#E3:50_1_1_2_2_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_1.html#T56">XBOOLE_1:56</a></i>;<br/>
<a NAME="E6:50_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="trees_1.html#E3:50_1_1_2_2_1_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:50_1_1_2_2_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_1_2_2_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:50_1_1_2_2_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><b>thus </b><a NAME="E4:50_1_1_2_2_2"/>
 <a href="trees_1.html#K1" target="_self">ProperPrefixes</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:50_1_1_2_2_2_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K1" target="_self">ProperPrefixes</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="trees_1.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E3:50_1_1_2_2_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="trees_1.html#E3:50_1_1_2_2_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="trees_1.html#E1:50_1_1_2_2_2_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="trees_1.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E4:50_1_1_2_2_2_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</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>10</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E6:50_1_1_2_2_2_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="trees_1.html#T8" target="_self">Th8</a></i>;<br/>
<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_2_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_1_2_2_2_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:50_1_1_2_2_2_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="trees_1.html#E6:50_1_1_2_2_2_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T64">FINSEQ_1:64</a></i>;<br/><a NAME="E4:50_1_1_2_2_2_1_1"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></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">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:50_1_1_2_2_2_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="trees_1.html#E6:50_1_1_2_2_2_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="trees_1.html#E3:50_1_1_2_2_2_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E6:50_1_1_2_2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><a NAME="E7:50_1_1_2_2_2_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T33">FINSEQ_1:33</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</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="trees_1.html#E4:50_1_1_2_2_2_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/><a NAME="E9:50_1_1_2_2_2_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="trees_1.html#E4:50_1_1_2_2_2_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="trees_1.html#E7:50_1_1_2_2_2_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="trees_1.html#D1" target="_self">Def1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="trees_1.html#T45" target="_self">Th45</a></i>;<br/><a NAME="E11:50_1_1_2_2_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="trees_1.html#E3:50_1_1_2_2_2_1_1"><i><font color="Green">E39</font></i></a></i>;<br/><a NAME="E12:50_1_1_2_2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E13:50_1_1_2_2_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_2_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_1_2_2_2_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:50_1_1_2_2_2_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="trees_1.html#E6:50_1_1_2_2_2_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T64">FINSEQ_1:64</a></i>;<br/><a NAME="E3:50_1_1_2_2_2_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T33">FINSEQ_1:33</a></i>;<br/><a NAME="E4:50_1_1_2_2_2_1_2"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</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="trees_1.html#E3:50_1_1_2_2_2_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/><a NAME="E6:50_1_1_2_2_2_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="trees_1.html#E3:50_1_1_2_2_2_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="trees_1.html#E4:50_1_1_2_2_2_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="trees_1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E7:50_1_1_2_2_2_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="trees_1.html#E1:50"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xboole_1.html#T116">XBOOLE_1:116</a>, <a class="ref" href="trees_1.html#T45" target="_self">Th45</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="trees_1.html#E1:50"><i><font color="Green">E31</font></i></a>, <a class="txt" href="trees_1.html#E6:50_1_1_2_2_2_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="trees_1.html#T45" target="_self">Th45</a></i>;<br/><a NAME="E9:50_1_1_2_2_2_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="trees_1.html#E7:50_1_1_2_2_2_1_2"><i><font color="Green">E42</font></i></a></i>;<br/><b>hence </b><a NAME="E10:50_1_1_2_2_2_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:50_1_1_2_2_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="trees_1.html#E7:50_1_1_2_2_2_1"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:50_1_1_2_2"/>
 <a href="trees_1.html#K1" target="_self">ProperPrefixes</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="trees_1.html#E1:50_1_1_2_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="trees_1.html#E2:50_1_1_2_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<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/><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><a NAME="E3:50_1_1_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_1_2_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/><a NAME="E2:50_1_1_2_3"/><b>then </b><i><font color="Green">E35</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="trees_1.html#E3:50_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="trees_1.html#D5" target="_self">Def5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_1_2_3_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:50_1_1_2_3_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <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="trees_1.html#T32" target="_self">Th32</a></i>;<br/><b>hence </b><a NAME="E3:50_1_1_2_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="trees_1.html#E2:50_1_1_2_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="trees_1.html#T31" target="_self">Th31</a></i>;<br/></div><b>end;</b></div><a NAME="E5:50_1_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E6:50_1_1_2_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_1_2_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:50_1_1_2_4"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 ;<br/><div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_1_2_4_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:50_1_1_2_4_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="trees_1.html#E3:50_1_1_2_4"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#T64">FINSEQ_1:64</a></i>;<br/><a NAME="E4:50_1_1_2_4_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="trees_1.html#E3:50_1_1_2_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="trees_1.html#E3:50_1_1_2_4_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E5:50_1_1_2_4_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<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="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_4_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_1_2_4_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>10</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>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="trees_1.html#E1:50"><i><font color="Green">E31</font></i></a>, <a class="txt" href="trees_1.html#E3:50_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="trees_1.html#E3:50_1_1_2_4_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="trees_1.html#D5" target="_self">Def5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_4_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_1_2_4_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E2:50_1_1_2_4_1_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/> <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><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></span>)</span>
<b>by </b><i><a class="txt" href="trees_1.html#E3:50_1_1_2_4_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="trees_1.html#E1:50_1_1_2_4_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/><b>hence </b><a NAME="E4:50_1_1_2_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="trees_1.html#E2:50_1_1_2_4_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="trees_1.html#T15" target="_self">Th15</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:50_1_1_2_4_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_4_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_1_2_4_1_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>9</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> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><a NAME="E2:50_1_1_2_4_1_2"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <a href="xboole_0.html#K1">{}</a>  <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><a NAME="E3:50_1_1_2_4_1_2"/><b>then </b>
<a href="xboole_0.html#K1">{}</a>  <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="trees_1.html#E3:50_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="trees_1.html#E1:50_1_1_2_4_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="trees_1.html#D5" target="_self">Def5</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><a NAME="E5:50_1_1_2_4_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="trees_1.html#E3:50_1_1_2_4_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="trees_1.html#E1:50_1_1_2_4_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><b>hence </b><a NAME="E6:50_1_1_2_4_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:50_1_1_2_4_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="trees_1.html#E5:50_1_1_2_4_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="trees_1.html#E6:50_1_1_2_4_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="trees_1.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_1_2_4_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:50_1_1_2_4_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="trees_1.html#E3:50_1_1_2_4"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#T64">FINSEQ_1:64</a></i>;<br/><a NAME="E4:50_1_1_2_4_2"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></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">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:50_1_1_2_4_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><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></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="trees_1.html#E3:50_1_1_2_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="trees_1.html#E3:50_1_1_2_4_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E6:50_1_1_2_4_2"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K7">^</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="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><a NAME="E7:50_1_1_2_4_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T33">FINSEQ_1:33</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</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="trees_1.html#E4:50_1_1_2_4_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="trees_1.html#E3:50_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="trees_1.html#E6:50_1_1_2_4_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="trees_1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E10:50_1_1_2_4_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="trees_1.html#E3:50_1_1_2_4_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E11:50_1_1_2_4_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E12:50_1_1_2_4_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:50_1_1_2_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="trees_1.html#E4:50_1_1_2_4"><i><font color="Green">E36</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:50_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="trees_1.html#E3:50_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="trees_1.html#E4:50_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="trees_1.html#NM1">Tree</a> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>thus </b><a NAME="E8:50_1_1"/>
(  not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> or ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>7</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>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_1_1_3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_1_3_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/><a NAME="E2:50_1_1_3_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><b>hence </b><a NAME="E3:50_1_1_3_1"/>
( ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>7</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>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_1_3_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/><a NAME="E2:50_1_1_3_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><b>hence </b><a NAME="E3:50_1_1_3_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>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:50_1_1_3"/>
( ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>7</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>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="trees_1.html#E1:50_1_1_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="trees_1.html#E2:50_1_1_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>assume </b><a NAME="E9:50_1_1"/><i><font color="Green">E33</font></i>: 
( ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>7</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>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<a NAME="E10:50_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>7</sub></font> implies <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> : not <font color="Maroon">c<sub>2</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  )
 
;<br/>
<a NAME="E11:50_1_1"/>
( 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>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font> ) implies <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  )
 
;<br/>
<b>hence </b><a NAME="E12:50_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="trees_1.html#E9:50_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="trees_1.html#E10:50_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div>
