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<b>let </b><font color="Maroon" title="c1">D1</font>, <font color="Maroon" title="c2">D2</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">d1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D1</font>;<br/><b>let </b><font color="Maroon" title="c4">d2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D2</font>;<br/><b>let </b><font color="Maroon" title="c5">F</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="trees_3.html#NM8" title="TREES_3:NM.8">DTree-set</a> of <font color="Maroon" title="c1">D1</font>,<font color="Maroon" title="c2">D2</font>;<br/><b>let </b><font color="Maroon" title="c6">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c5">F</font>;<br/>











<a NAME="E1:52"/><i><font color="Green" title="E22">A1</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">p</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">T</font> being   <a href="trees_3.html#M5" title="TREES_3:mode.5">Element</a> of <font color="Maroon" title="c5">F</font> st <br/>( <font color="Olive" title="b1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">T</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a>  );<br/>
<a NAME="E2:52"/><i><font color="Green" title="E23">A2</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="trees_3.html#M5" title="TREES_3:mode.5">Element</a> of  <a href="trees_3.html#K4" title="TREES_3:func.4">Trees</a> <font color="Maroon" title="c2">D2</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:52_1"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">T</font> = <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> as    <a href="trees_3.html#M5" title="TREES_3:mode.5">Element</a> of <font color="Maroon" title="c5">F</font> <b>by </b><i><a class="txt" href="trees_4.html#E1:52"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="trees_4.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c8">T</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a>  as    <a href="trees_3.html#M5" title="TREES_3:mode.5">Element</a> of  <a href="trees_3.html#K4" title="TREES_3:func.4">Trees</a> <font color="Maroon" title="c2">D2</font> <b>by </b><i><a class="ref" href="trees_3.html#D7" title="TREES_3:def.7">TREES_3:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">y</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c8">T</font>
;<br/>

<b>thus </b><a NAME="E4:52_1"/>
( <font color="Maroon" title="c8">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">T</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a>  )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c7">p2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="trees_3.html#K4" title="TREES_3:func.4">Trees</a> <font color="Maroon" title="c2">D2</font><b> such that </b><br/><a NAME="E4:52"/><i><font color="Green" title="E24">A3</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font></span>)</span> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c7">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="trees_4.html#E2:52"><i><font color="Green" title="E23">A2</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c7">p2</font>
;<br/>

<b>thus </b><a NAME="E5:52"/><i><font color="Green" title="E25">A4</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="txt" href="trees_4.html#E4:52"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>

<b>hence </b><a NAME="E6:52"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">p</font> holds <br/> ex <font color="Olive" title="b2">T</font> being   <a href="trees_3.html#M5" title="TREES_3:mode.5">Element</a> of <font color="Maroon" title="c5">F</font> st <br/>( <font color="Olive" title="b2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c7">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">T</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a>  )
 <b>by </b><i><a class="txt" href="trees_4.html#E4:52"><i><font color="Green" title="E24">A3</font></i></a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:52_2"/>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">p</font>
 ;<br/><a NAME="E2:52_2"/><b>then </b>
 ex <font color="Olive" title="b1">T</font> being   <a href="trees_3.html#M5" title="TREES_3:mode.5">Element</a> of <font color="Maroon" title="c5">F</font> st <br/>( <font color="Olive" title="b1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c7">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">T</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a>  )
 <b>by </b><i><a class="txt" href="trees_4.html#E4:52"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="trees_4.html#E5:52"><i><font color="Green" title="E25">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E3:52_2"/>
for <font color="Olive" title="b1">T</font> being  <a href="trees_3.html#NM7" title="TREES_3:NM.7">DecoratedTree</a> of <font color="Maroon" title="c1">D1</font>,<font color="Maroon" title="c2">D2</font>  st <font color="Olive" title="b1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> holds <br/><font color="Maroon" title="c7">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">T</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a> 
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:52"/>
<span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">d2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d2</font> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c7">p2</font>
 <b>by </b><i><a class="txt" href="trees_4.html#E5:52"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="trees_4.html#T28" target="_self" title="TREES_4:th.28">Th28</a></i>;<br/>


</div>
