<?xml version="1.0"?>
<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() implies <font color="Maroon">P<sub>1</sub></font>[$1] );<br/>
<a NAME="E1:7_1"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<b>by </b><i><a class="txt" href="hilbert2.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a></i>;<br/>
<a NAME="E2:7_1"/><i><font color="Green" title="E9">A4</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</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="Olive" title="b2">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:7_1_1"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>]
 ;<br/>

<b>assume </b><a NAME="E2:7_1_1"/><i><font color="Green" title="E11">A6</font></i>: 
<font color="Maroon" title="c1">p</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="c2">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">h</font> = <font color="Maroon" title="c1">p</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="c2">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as    <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>  <b>by </b><i><a class="ref" href="trees_1.html#T44" title="TREES_1:th.44">TREES_1:44</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">g</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> , <font color="Maroon" title="c5">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> <b> such that </b><br/><a NAME="E5:7_1_1"/><i><font color="Green" title="E12">A7</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="hilbert2.html#T4" target="_self" title="HILBERT2:th.4">Th4</a></i>;<br/>
<a NAME="E6:7_1_1"/><i><font color="Green" title="E13">A8</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="hilbert2.html#E5:7_1_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="finseq_2.html#T20" title="FINSEQ_2:th.20">FINSEQ_2:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c4">g</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="hilbert2.html#E2:7_1_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="hilbert2.html#E5:7_1_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="trees_1.html#T46" title="TREES_1:th.46">TREES_1:46</a></i>;<br/>
<a NAME="E8:7_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">g</font>]
 
<b>by </b><i><a class="txt" href="hilbert2.html#E1:7_1_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="hilbert2.html#E6:7_1_1"><i><font color="Green" title="E13">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:7_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">p</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="c2">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>]
 <b>by </b><i><a class="txt" href="hilbert2.html#E2:7"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="hilbert2.html#E2:7_1_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="hilbert2.html#E5:7_1_1"><i><font color="Green" title="E12">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:7_1"/>
for <font color="Olive" title="b1">p</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>]
 
<b>from </b><i><a class="ref" href="finseq_1.html#S3" title="FINSEQ_1:sch.3">FINSEQ_1:sch 3</a>(<a class="txt" href="hilbert2.html#E1:7_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="hilbert2.html#E2:7_1"><i><font color="Green" title="E9">A4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:7_1"/>
for <font color="Olive" title="b1">f</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">f</font>]
 ;<br/>


</div>
