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

<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">F</font><br/> for <font color="Olive" title="b2">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="b2">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="Olive" title="b1">A</font> &amp; <font color="Olive" title="b1">A</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a>  holds <br/> ex <font color="Olive" title="b3">G</font> being   <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">i</font> st <font color="Olive" title="b3">G</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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="c2">A</font> &amp; <font color="Maroon" title="c2">A</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a>  holds <br/> ex <font color="Olive" title="b2">G</font> being   <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">i</font> st <font color="Olive" title="b2">G</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">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="c2">A</font> &amp; <font color="Maroon" title="c2">A</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a>  implies  ex <font color="Olive" title="b1">G</font> being   <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font> st <font color="Olive" title="b1">G</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:45"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<font color="Maroon" title="c3">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="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E2:45"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">G</font> being   <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font> st <font color="Olive" title="b1">G</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> </span><br/>

<a NAME="E3:45"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="hallmar1.html#T16" target="_self" title="HALLMAR1:th.16">Th16</a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">n</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span>;<br/>
<a NAME="E4:45"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="hallmar1.html#T16" target="_self" title="HALLMAR1:th.16">Th16</a></span>;<br/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span> ex <font color="Olive" title="b1">G</font> being   <a href="hallmar1.html#M2" title="HALLMAR1:mode.2">Reduction</a> of <font color="Maroon" title="c2">A</font> st <br/>( <font color="Olive" title="b1">G</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 );<br/>
<a NAME="E5:45"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
<font color="Maroon" title="c2">A</font> is    <a href="hallmar1.html#M2" title="HALLMAR1:mode.2">Reduction</a> of <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hallmar1.html#T26" target="_self" title="HALLMAR1:th.26">Th26</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:45_1"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1"><span class="kw">suppose </span></a><a NAME="E1:45_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">G</font> being   <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font> st <font color="Olive" title="b1">G</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> </span><br/><div class="add"><a NAME="E2:45_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> is    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="ref" href="hallmar1.html#D7" target="_self" title="HALLMAR1:def.7">Def7</a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></span>;<br/><span class="kw">hence </span><a NAME="E3:45_1_1"/>
 ex <font color="Olive" title="b1">G</font> being   <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font> st <font color="Olive" title="b1">G</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E37">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2"><span class="kw">suppose </span></a><a NAME="E1:45_1_2"/><span class="lab"><font color="Green" title="E41">A6</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">G</font> being   <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font> st <font color="Olive" title="b1">G</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> </span><br/><div class="add"><a NAME="E2:45_1_2"/><span class="lab"><font color="Green" title="E42">A7</font></span>: 
for <font color="Olive" title="b1">k</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">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">k</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:45_1_2_1"/><span class="lab"><font color="Green" title="E43">A8</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">and </span><br/><a NAME="E2:45_1_2_1"/><span class="lab"><font color="Green" title="E44">A9</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">G</font> being    <a href="hallmar1.html#M2" title="HALLMAR1:mode.2">Reduction</a> of <font color="Maroon" title="c2">A</font><span class="kw"> such that </span><br/><a NAME="E4:45_1_2_1"/><span class="lab"><font color="Green" title="E45">A10</font></span>: 
<font color="Maroon" title="c6">G</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> 
 <span class="kw">and </span><br/><a NAME="E5:45_1_2_1"/><span class="lab"><font color="Green" title="E46">A11</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_2_1"><span class="lab"><font color="Green" title="E44">A9</font></span></a></span>;<br/>
<a NAME="E6:45_1_2_1"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_2_1"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:45_1_2_1"/><span class="lab"><font color="Green" title="E47">A12</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">and </span><br/><a NAME="E9:45_1_2_1"/><span class="lab"><font color="Green" title="E48">A13</font></span>: 
 <a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c7">x</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_1_2_1"><span class="lab"><font color="Green" title="E45">A10</font></span></a>, <a class="txt" href="#E5:45_1_2_1"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="ref" href="hallmar1.html#T29" target="_self" title="HALLMAR1:th.29">Th29</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">H</font> =  <a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c7">x</font>;<br/>
<a NAME="E10:45_1_2_1"/><span class="lab"><font color="Green" title="E49">A14</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">G</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="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hallmar1.html#D6" target="_self" title="HALLMAR1:def.6">Def6</a></span>;<br/>
<a NAME="E11:45_1_2_1"/><span class="kw">then </span>
 <a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c7">x</font> is    <a href="hallmar1.html#M2" title="HALLMAR1:mode.2">Reduction</a> of <font color="Maroon" title="c6">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="hallmar1.html#T21" target="_self" title="HALLMAR1:th.21">Th21</a></span>;<br/>
<a NAME="E12:45_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A15</font></span>: 
 <a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c7">x</font> is    <a href="hallmar1.html#M2" title="HALLMAR1:mode.2">Reduction</a> of <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hallmar1.html#T22" target="_self" title="HALLMAR1:th.22">Th22</a></span>;<br/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c7">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E5:45_1_2_1"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="txt" href="#E10:45_1_2_1"><span class="lab"><font color="Green" title="E49">A14</font></span></a>, <a class="txt" href="#E8:45_1_2_1"><span class="lab"><font color="Green" title="E47">A12</font></span></a>, <a class="ref" href="hallmar1.html#T11" target="_self" title="HALLMAR1:th.11">Th11</a></span>
<br/>.= 
<font color="Maroon" title="c5">k</font>

;<br/>
<span class="kw">hence </span><a NAME="E14:45_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_1_2_1"><span class="lab"><font color="Green" title="E48">A13</font></span></a>, <a class="txt" href="#E12:45_1_2_1"><span class="lab"><font color="Green" title="E50">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:45_1_2"/>
<font color="Maroon" title="c2">A</font> is    <a href="hallmar1.html#M2" title="HALLMAR1:mode.2">Reduction</a> of <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hallmar1.html#T26" target="_self" title="HALLMAR1:th.26">Th26</a></span>;<br/><a NAME="E4:45_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A16</font></span>: 
 ex <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 <br/>( <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E1:45_1_2"><span class="lab"><font color="Green" title="E41">A6</font></span></a></span>;<br/><a NAME="E5:45_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[1]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="hallmar1.html#S2" target="_self" title="HALLMAR1:sch.2">HALLMAR1:sch 2</a>(<a class="txt" href="#E4:45_1_2"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E2:45_1_2"><span class="lab"><font color="Green" title="E42">A7</font></span></a>);<br/></span><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">G</font> being    <a href="hallmar1.html#M2" title="HALLMAR1:mode.2">Reduction</a> of <font color="Maroon" title="c2">A</font><span class="kw"> such that </span><br/><a NAME="E7:45_1_2"/><span class="lab"><font color="Green" title="E44">A17</font></span>: 
<font color="Maroon" title="c5">G</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> 
 <span class="kw">and </span><br/><a NAME="E8:45_1_2"/><span class="lab"><font color="Green" title="E45">A18</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E9:45_1_2"/>
<font color="Maroon" title="c5">G</font> is    <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E8:45_1_2"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="ref" href="hallmar1.html#D7" target="_self" title="HALLMAR1:def.7">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E10:45_1_2"/>
 ex <font color="Olive" title="b1">G</font> being   <a href="hallmar1.html#M3" title="HALLMAR1:mode.3">Singlification</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font> st <font color="Olive" title="b1">G</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_1_2"><span class="lab"><font color="Green" title="E44">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
