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<b>let </b><font color="Maroon">D1</font> be   <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">D2</font> be   <a href="trees_2.html#NM4">DecoratedTree</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:63_1_2"/><i><font color="Green">E65</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">D1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font></span>)</span>
 <b>and </b><br/><a NAME="E2:63_1_2"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   holds <br/>(  not <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font></span>)</span> or ( not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">q</font> &amp; <font color="Maroon">D1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> ) or  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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font> &amp; <font color="Olive">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> &amp; <font color="Maroon">D1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> ) )
 <b>and </b><br/><a NAME="E3:63_1_2"/><i><font color="Green">E90</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">D2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font></span>)</span>
 <b>and </b><br/><a NAME="E4:63_1_2"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   holds <br/>(  not <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font></span>)</span> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font></span>)</span> or ( not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">q</font> &amp; <font color="Maroon">D2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> ) or  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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font> &amp; <font color="Olive">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> &amp; <font color="Maroon">D2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_2_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>assume </b><a NAME="E1:63_1_2_1"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> )
 ;<br/><a NAME="E2:63_1_2_1"/><b>then </b><i><font color="Green">E93</font></i>: 
( ( not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">q</font> &amp; <font color="Maroon">D1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> ) or  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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</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> &amp; <font color="Maroon">D1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> ) )
 <b>by </b><i>, </i>;<br/><a NAME="E3:63_1_2_1"/>
( ( not <font color="Maroon">p</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">q</font> &amp; <font color="Maroon">D2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> ) or  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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</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> &amp; <font color="Maroon">D2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> ) )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:63_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a>, <a class="ref" href="trees_1.html#T8">TREES_1:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:63_1_2"/>
<font color="Maroon">D1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font>
 <b>by </b><i>, , </i>;<br/>


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