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<b>let </b><font color="Maroon" title="c1">p</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="c2">T</font>, <font color="Maroon" title="c3">T1</font> be   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a>;<br/>





<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">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>assume </b><a NAME="E2:19"/><i><font color="Green" title="E18">A2</font></i>: 
( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">T1</font></span>)</span> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">s</font></span>)</span> where <font color="Olive" title="b1">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T1</font> : <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> </span>}</span>  )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1"/>
( ( not <font color="Maroon" title="c1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c4">q</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> ) or  ex <font color="Olive" title="b1">r</font> being   <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>  st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T1</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">r</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> ) )
 </span><b>by </b><i><a class="txt" href="trees_a.html#E1:19"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="trees_a.html#E2:19"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="trees_a.html#T14" target="_self" title="TREES_A:th.14">Th14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1"><b>suppose </b></a><a NAME="E1:19_1_1"/><i><font color="Green" title="E19">A3</font></i>: 
( not <font color="Maroon" title="c1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c4">q</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c5">r</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T1</font><b> such that </b><br/><a NAME="E3:19_1_1"/><i><font color="Green" title="E20">A4</font></i>: 
( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> )
 <b>by </b><i><a class="txt" href="trees_a.html#E2:19"><i><font color="Green" title="E18">A2</font></i></a></i>;<br/><b>thus </b><a NAME="E4:19_1_1"/>
 ex <font color="Olive" title="b1">r</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T1</font> st <br/>( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">r</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> )
 <b>by </b><i><a class="txt" href="trees_a.html#E1:19_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="trees_a.html#E3:19_1_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="trees_1.html#T8" title="TREES_1:th.8">TREES_1:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2"><b>suppose </b></a><a NAME="E1:19_1_2"/>
 ex <font color="Olive" title="b1">r</font> being   <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>  st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T1</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">r</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_1_2"/>
 ex <font color="Olive" title="b1">r</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T1</font> st <br/>( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">r</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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