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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">W1</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">W2</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>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>assume </b><a NAME="E1:11"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; not <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="xboole_0.html#R3">are_c=-comparable</a>  )
 ;<br/>

<a NAME="E2:11"/><b>then </b>
( not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">q</font> &amp; not <font color="Maroon">q</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">p</font> &amp; not <font color="Maroon">q</font>,<font color="Maroon">p</font> <a href="xboole_0.html#R3">are_c=-comparable</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>
<a NAME="E3:11"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">p</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">q</font>,<font color="Maroon">W2</font> &amp; <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>,<font color="Maroon">W1</font> )
 
<b>by </b><i>, </i>;<br/>
<b>let </b><font color="Maroon">r</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" target="_self">TREES_2:def 1</a><br/>

<b>thus </b><a NAME="E4:11"/>
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<font color="Maroon">W1</font></span>)</span> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">q</font>,<font color="Maroon">W2</font> implies <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">q</font>,<font color="Maroon">W2</font></span>)</span> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<font color="Maroon">W1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<font color="Maroon">W1</font></span>)</span> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">q</font>,<font color="Maroon">W2</font>
 ;<br/>

<a NAME="E2:11_1"/><b>then </b>
( ( <font color="Maroon">r</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>,<font color="Maroon">W1</font> &amp; not <font color="Maroon">q</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">r</font> ) or  ex <font color="Olive">r1</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r1</font> ) )
 
<b>by </b><i>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>
<a NAME="E3:11_1"/><b>then </b>
( ( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; not <font color="Maroon">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">r</font> &amp; not <font color="Maroon">q</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">r</font> ) or (  ex <font color="Olive">r2</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r2</font> ) &amp; not <font color="Maroon">q</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">r</font> ) or ( <font color="Maroon">q</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">r</font> &amp;  ex <font color="Olive">r1</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r1</font> ) ) )
 
<b>by </b><i>, <a class="ref" href="trees_1.html#T8">TREES_1:8</a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>
<a NAME="E4:11_1"/><b>then </b>
( ( <font color="Maroon">r</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">q</font>,<font color="Maroon">W2</font> &amp; not <font color="Maroon">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">r</font> ) or  ex <font color="Olive">r1</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r1</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="E5:11_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">q</font>,<font color="Maroon">W2</font></span>)</span> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<font color="Maroon">W1</font>
 <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="E5:11"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">q</font>,<font color="Maroon">W2</font></span>)</span> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<font color="Maroon">W1</font>
 ;<br/>

<a NAME="E6:11"/><b>then </b>
( ( <font color="Maroon">r</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">q</font>,<font color="Maroon">W2</font> &amp; not <font color="Maroon">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">r</font> ) or  ex <font color="Olive">r1</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r1</font> ) )
 
<b>by </b><i>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>
<a NAME="E7:11"/><b>then </b>
( ( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; not <font color="Maroon">q</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">r</font> &amp; not <font color="Maroon">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">r</font> ) or (  ex <font color="Olive">r2</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r2</font> ) &amp; not <font color="Maroon">p</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">r</font> ) or ( <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">r</font> &amp;  ex <font color="Olive">r1</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r1</font> ) ) )
 
<b>by </b><i>, <a class="ref" href="trees_1.html#T8">TREES_1:8</a>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>
<a NAME="E8:11"/><b>then </b>
( ( <font color="Maroon">r</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>,<font color="Maroon">W1</font> &amp; not <font color="Maroon">q</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">r</font> ) or  ex <font color="Olive">r1</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r1</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="E9:11"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<font color="Maroon">W1</font></span>)</span> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">q</font>,<font color="Maroon">W2</font>
 <b>by </b><i>, <a class="ref" href="trees_1.html#D12">TREES_1:def 12</a></i>;<br/>


</div>
