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<b>let </b><font color="Maroon">W</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:9"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><a NAME="E1:9_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> implies <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1"/>
<font color="Maroon">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">q</font>
 ;<br/><a NAME="E2:9_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E4:9_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="trees_1.html#T8">TREES_1:8</a></i>;<br/><a NAME="E5:9_1_1_1"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">q</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T43">FINSEQ_1:43</a></i>;<br/><a NAME="E6:9_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>take </b><font color="Maroon">r</font> = <font color="Maroon">r</font>;<br/><b>thus </b><a NAME="E8:9_1_1_1"/>
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font> )
 <b>by </b><i>, , , <a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:9_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, , <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="E2:9_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font></span>)</span> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 ;<br/><a NAME="E3:9_1"/><b>then </b>
 ex <font color="Olive">r</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r</font> )
 <b>by </b><i>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/><b>hence </b><a NAME="E4:9_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:9"/>
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i/>;<br/>


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