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<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_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_2.html#D1">TREES_2:def 1</a><br/>







<b>thus </b><a NAME="E1:12"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_a.html#K1" target="_self">tree</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_a.html#K1" target="_self">tree</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><i><font color="Green">E11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_1_1"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( 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> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span> holds <br/> not <font color="Olive">b<sub>1</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="E2:12_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="trees_a.html#E1:12_1_1"><i><font color="Green">E12</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>5</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:12_1_2"/><i><font color="Green">E12</font></i>: 
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="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><a NAME="E2:12_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="trees_a.html#E1:12_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_1_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>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</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_a.html#E1:12_1_2"><i><font color="Green">E12</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:12_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="trees_a.html#E1:12_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="trees_a.html#E2:12_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="trees_a.html#D1" target="_self">Def1</a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>


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

<b>assume </b><a NAME="E2:12"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:12"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>4</sub></font> implies ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( 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> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span> holds <br/> not <font color="Olive">b<sub>1</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>4</sub></font> ) ) )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2"/><i><font color="Green">E12</font></i>: 
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>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><b>thus </b><a NAME="E2:12_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<a NAME="E1:12_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:12_2_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="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</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_a.html#E1:12_2"><i><font color="Green">E12</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:12"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_a.html#K1" target="_self">tree</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="trees_a.html#E2:12"><i><font color="Green">E10</font></i></a>, <a class="txt" href="trees_a.html#E3:12"><i><font color="Green">E11</font></i></a>, <a class="ref" href="trees_a.html#D1" target="_self">Def1</a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>


</div>
