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<b>let </b><font color="Maroon">F</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F</font>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:45"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> is <a href="hallmar1.html#V1" target="_self">Hall</a> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">G</font> being   <a href="hallmar1.html#M2" target="_self">Reduction</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">G</font> is <a href="hallmar1.html#V1" target="_self">Hall</a> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Olive">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<b>set </b><font color="Maroon">n</font> =  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>;<br/>
<a NAME="E2:45"/><i><font color="Green">E24</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:45"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="card_1.html#T78">CARD_1:78</a></i>;<br/>
<a NAME="E4:45"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">A</font> is    <a href="hallmar1.html#M2" target="_self">Reduction</a> of <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> 1 or  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="hallmar1.html#T2" target="_self">Th2</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1"><b>suppose </b></a><a NAME="E1:45_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:45_1_1"/><b>then </b>
<font color="Maroon">A</font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">A</font>,<font color="Maroon">i</font>
 <b>by </b><i>, , <a class="txt" href="hallmar1.html#E2:45"><i><font color="Green">E24</font></i></a>, <a class="ref" href="card_1.html#T78">CARD_1:78</a></i>;<br/><b>hence </b><a NAME="E3:45_1_1"/>
 ex <font color="Olive">G</font> being   <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">A</font>,<font color="Maroon">i</font> st <font color="Olive">G</font> is <a href="hallmar1.html#V1" target="_self">Hall</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2"><b>suppose </b></a><a NAME="E1:45_1_2"/><i><font color="Green">E28</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:45_1_2"/>
( <font color="Maroon">A</font> is    <a href="hallmar1.html#M2" target="_self">Reduction</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> is <a href="hallmar1.html#V1" target="_self">Hall</a> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="hallmar1.html#E4:45_1_2"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E3:45_1_2"/><b>then </b><i><font color="Green">E29</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] )
 <b>by </b><i/>;<br/><a NAME="E4:45_1_2"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_2_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1] )
 ;<br/>

<b>then consider </b><font color="Maroon">G</font> being    <a href="hallmar1.html#M2" target="_self">Reduction</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:45_1_2_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">G</font> is <a href="hallmar1.html#V1" target="_self">Hall</a> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>
<a NAME="E4:45_1_2_1"/><i><font color="Green">E72</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E1:45"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E5:45_1_2_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:45_1_2_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp;  <a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">G</font>,<font color="Maroon">i</font>,<font color="Maroon">x</font> is <a href="hallmar1.html#V1" target="_self">Hall</a> )
 <b>by </b><i><a class="ref" href="hallmar1.html#T3" target="_self">Th3</a>, </i>;<br/>
<b>set </b><font color="Maroon">H</font> =  <a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">G</font>,<font color="Maroon">i</font>,<font color="Maroon">x</font>;<br/>
<a NAME="E8:45_1_2_1"/>
 <a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">G</font>,<font color="Maroon">i</font>,<font color="Maroon">x</font> is    <a href="hallmar1.html#M2" target="_self">Reduction</a> of <font color="Maroon">G</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E9:45_1_2_1"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">G</font>,<font color="Maroon">i</font>,<font color="Maroon">x</font> is    <a href="hallmar1.html#M2" target="_self">Reduction</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E1:45_1_2"><i><font color="Green">E28</font></i></a></i>;<br/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">G</font>,<font color="Maroon">i</font>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i>, <a class="ref" href="hallmar1.html#T3" target="_self">Th3</a>, , , </i>
<br/>.= 
<font color="Maroon">k</font>

;<br/>
<b>hence </b><a NAME="E11:45_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 <b>by </b><i>, <a class="ref" href="hallmar1.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:45_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[1]
 <b>from </b><i><a class="ref" href="hallmar1.html#S2" target="_self">HALLMAR1:sch 2</a>(, <a class="ref" href="hallmar1.html#D1" target="_self">Def1</a>);<br/></i><b>then consider </b><font color="Maroon">G</font> being    <a href="hallmar1.html#M2" target="_self">Reduction</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E7:45_1_2"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">G</font> is <a href="hallmar1.html#V1" target="_self">Hall</a> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><a NAME="E8:45_1_2"/>
<font color="Maroon">G</font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">A</font>,<font color="Maroon">i</font>
 <b>by </b><i>, , , <a class="txt" href="hallmar1.html#E2:45"><i><font color="Green">E24</font></i></a></i>;<br/><b>hence </b><a NAME="E9:45_1_2"/>
 ex <font color="Olive">G</font> being   <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">A</font>,<font color="Maroon">i</font> st <font color="Olive">G</font> is <a href="hallmar1.html#V1" target="_self">Hall</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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