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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">p</font> being <a href="trees_3.html#V5" title="TREES_3:attr.5">FinTree-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> holds <br/> <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <font color="Olive" title="b1">p</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>;<br/>
<a NAME="E1:126_1_1"/><i><font color="Green" title="E53">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="trees_3.html#T55" target="_self" title="TREES_3:th.55">Th55</a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E2:126_1_1"/><i><font color="Green" title="E54">A2</font></i>: 
for <font color="Olive" title="b1">n</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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="126_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">n</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="E1:126_1_1_1"/><i><font color="Green" title="E55">A3</font></i>: 
for <font color="Olive" title="b1">p</font> being <a href="trees_3.html#V5" title="TREES_3:attr.5">FinTree-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> holds <br/> <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <font color="Olive" title="b1">p</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">p</font> be  <a href="trees_3.html#V5" title="TREES_3:attr.5">FinTree-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E2:126_1_1_1"/><i><font color="Green" title="E56">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/>

<a NAME="E3:126_1_1_1"/>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="trees_3.html#E2:126_1_1_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">q</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>, <font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:126_1_1_1"/><i><font color="Green" title="E57">A5</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T63" title="FINSEQ_1:th.63">FINSEQ_1:63</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">q</font> = <font color="Maroon" title="c4">q</font> as  <a href="trees_3.html#V5" title="TREES_3:attr.5">FinTree-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="txt" href="trees_3.html#E5:126_1_1_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="ref" href="trees_3.html#T28" target="_self" title="TREES_3:th.28">Th28</a></i>;<br/>
<a NAME="E7:126_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E8:126_1_1_1"/><b>then </b><i><font color="Green" title="E58">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="trees_3.html#E5:126_1_1_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E9:126_1_1_1"/><b>then </b>
 <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <font color="Maroon" title="c6">q</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="trees_3.html#E1:126_1_1_1"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="trees_3.html#E2:126_1_1_1"><i><font color="Green" title="E56">A4</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">W</font> = <span class="p1">(<span class="default"><a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> <b>by </b><i><a class="ref" href="trees_1.html#T52" title="TREES_1:th.52">TREES_1:52</a></i>;<br/>
<a NAME="E11:126_1_1_1"/>
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> is <a href="trees_3.html#V5" title="TREES_3:attr.5">FinTree-yielding</a>
 
<b>by </b><i><a class="txt" href="trees_3.html#E5:126_1_1_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="ref" href="trees_3.html#T28" target="_self" title="TREES_3:th.28">Th28</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c5">x</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> <b>by </b><i><a class="ref" href="trees_3.html#T31" target="_self" title="TREES_3:th.31">Th31</a></i>;<br/>
<a NAME="E13:126_1_1_1"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E14:126_1_1_1"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="trees_1.html#T55" title="TREES_1:th.55">TREES_1:55</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">r</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c7">W</font> <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E16:126_1_1_1"/>
 <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="txt" href="trees_3.html#E2:126_1_1_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="trees_3.html#E5:126_1_1_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="txt" href="trees_3.html#E8:126_1_1_1"><i><font color="Green" title="E58">A6</font></i></a>, <a class="ref" href="trees_3.html#T58" target="_self" title="TREES_3:th.58">Th58</a></i>;<br/>
<b>hence </b><a NAME="E17:126_1_1_1"/>
 <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <font color="Maroon" title="c3">p</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:126_1_1"/>
for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="trees_3.html#E1:126_1_1"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="trees_3.html#E2:126_1_1"><i><font color="Green" title="E54">A2</font></i></a>);<br/></i>
<a NAME="E4:126_1_1"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font> implies  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <font color="Maroon" title="c1">p</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
;<br/>
<b>hence </b><a NAME="E5:126_1_1"/>
 <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <font color="Maroon" title="c1">p</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>


</div>
