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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="finseq_1.html#NM1">FinSequence</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>] );<br/>
<a NAME="E1:7_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<b>by </b><i><a class="txt" href="hilbert2.html#E1:7"><i><font color="Green">E6</font></i></a></i>;<br/>
<a NAME="E2:7_1"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="finseq_1.html#K9">*&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">c<sub>1</sub></font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



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

<b>assume </b><a NAME="E2:7_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="trees_1.html#T44">TREES_1:44</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:7_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="hilbert2.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E6:7_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="hilbert2.html#E5:7_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_2.html#T20">FINSEQ_2:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="trees_1.html#M1">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">E11</font></i></a>, <a class="txt" href="hilbert2.html#E5:7_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="trees_1.html#T46">TREES_1:46</a></i>;<br/>
<a NAME="E8:7_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>]
 
<b>by </b><i><a class="txt" href="hilbert2.html#E1:7_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="hilbert2.html#E6:7_1_1"><i><font color="Green">E13</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">c<sub>1</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span>]
 <b>by </b><i><a class="txt" href="hilbert2.html#E2:7"><i><font color="Green">E7</font></i></a>, <a class="txt" href="hilbert2.html#E2:7_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="hilbert2.html#E5:7_1_1"><i><font color="Green">E12</font></i></a></i>;<br/>


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


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