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<b>let </b><font color="Maroon" title="c1">W</font> be   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c4">C</font> be    <a href="trees_2.html#M1" title="TREES_2:mode.1">Chain</a> of <font color="Maroon" title="c1">W</font>;<br/>







<b>assume </b><a NAME="E1:41"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font> &amp; not <font color="Maroon" title="c2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c3">q</font> )
 ;<br/>

<a NAME="E2:41"/><b>then </b>
( <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K1" title="TREES_1:func.1">ProperPrefixes</a> <font color="Maroon" title="c2">p</font> &amp; not <font color="Maroon" title="c3">q</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c2">p</font> )
 
<b>by </b><i><a class="ref" href="trees_2.html#T23" target="_self" title="TREES_2:th.23">Th23</a>, <a class="ref" href="trees_1.html#T24" title="TREES_1:th.24">TREES_1:24</a></i>;<br/>
<b>hence </b><a NAME="E3:41"/>
contradiction
 <b>by </b><i><a class="ref" href="trees_1.html#T36" title="TREES_1:th.36">TREES_1:36</a></i>;<br/>


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