<?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#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b2">i</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI: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#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">i</font> <a href="tarski.html#R2" title="TARSKI: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="E1:47"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c3">i</font> <a href="tarski.html#R2" title="TARSKI: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="E2:47"/><span class="lab"><font color="Green" title="E35">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E3:47"/><span class="lab"><font color="Green" title="E36">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="#E1:47"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="hallmar1.html#T16" target="_self" onmouseover="rs('hallmar1/T16')" onmouseout="rh()">Th16</a>, <a class="ref" href="card_1.html#T27" onmouseover="rs('card_1/T27')" onmouseout="rh()">CARD_1:27</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">n</font> =  <a href="card_1.html#K4" title="CARD_1:func.4">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="E4:47"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">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="#E1:47"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="hallmar1.html#T16" target="_self" onmouseover="rs('hallmar1/T16')" onmouseout="rh()">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#NK1" title="NUMBERS:NK.1">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#K4" title="CARD_1:func.4">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="E5:47"/><span class="lab"><font color="Green" title="E38">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" onmouseover="rs('hallmar1/T26')" onmouseout="rh()">Th26</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:47_1"/>
(  <a href="card_1.html#K4" title="CARD_1:func.4">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#K4" title="CARD_1:func.4">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="#E4:47"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1"><span class="kw">suppose </span></a><a NAME="E1:47_1_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E2:47_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="#E1:47"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E5:47"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="hallmar1.html#D7" target="_self" onmouseover="rs('hallmar1/D7')" onmouseout="rh()">Def7</a>, <a class="ref" href="card_1.html#T27" onmouseover="rs('card_1/T27')" onmouseout="rh()">CARD_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E3:47_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="#E2:47"><span class="lab"><font color="Green" title="E35">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="47_1_2"><span class="kw">suppose </span></a><a NAME="E1:47_1_2"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E2:47_1_2"/><span class="lab"><font color="Green" title="E40">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#NK1" title="NUMBERS:NK.1">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#K1" title="NAT_1:func.1">+</a> 1] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_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#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K1" title="NAT_1:func.1">+</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="E1:47_1_2_1"/><span class="lab"><font color="Green" title="E41">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="E2:47_1_2_1"/><span class="lab"><font color="Green" title="E42">A9</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E4:47_1_2_1"/><span class="lab"><font color="Green" title="E43">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="E5:47_1_2_1"/><span class="lab"><font color="Green" title="E44">A11</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">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#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1_2_1"><span class="lab"><font color="Green" title="E42">A9</font></span></a></span>;<br/>
<a NAME="E6:47_1_2_1"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_2_1"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</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#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E8:47_1_2_1"/><span class="lab"><font color="Green" title="E45">A12</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI: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="E9:47_1_2_1"/><span class="lab"><font color="Green" title="E46">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="#E4:47_1_2_1"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E5:47_1_2_1"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="ref" href="hallmar1.html#T29" target="_self" onmouseover="rs('hallmar1/T29')" onmouseout="rh()">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="E10:47_1_2_1"/><span class="lab"><font color="Green" title="E47">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" onmouseover="rs('hallmar1/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E11:47_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="#E1:47"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="hallmar1.html#T21" target="_self" onmouseover="rs('hallmar1/T21')" onmouseout="rh()">Th21</a></span>;<br/>
<a NAME="E12:47_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">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" onmouseover="rs('hallmar1/T22')" onmouseout="rh()">Th22</a></span>;<br/>
 <a href="card_1.html#K4" title="CARD_1:func.4">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#K1" title="NAT_1:func.1">+</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="#E1:47"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E5:47_1_2_1"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="txt" href="#E10:47_1_2_1"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="txt" href="#E8:47_1_2_1"><span class="lab"><font color="Green" title="E45">A12</font></span></a>, <a class="ref" href="hallmar1.html#T11" target="_self" onmouseover="rs('hallmar1/T11')" onmouseout="rh()">Th11</a></span>
<br/>.= 
<font color="Maroon" title="c5">k</font>

;<br/>
<span class="kw">hence </span><a NAME="E14:47_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="#E9:47_1_2_1"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E12:47_1_2_1"><span class="lab"><font color="Green" title="E48">A15</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:47_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" onmouseover="rs('hallmar1/T26')" onmouseout="rh()">Th26</a></span>;<br/><a NAME="E4:47_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">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#NK1" title="NUMBERS:NK.1">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="#E2:47"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E1:47_1_2"><span class="lab"><font color="Green" title="E39">A6</font></span></a></span>;<br/><a NAME="E5:47_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" onmouseover="rs('hallmar1/S2')" onmouseout="rh()">HALLMAR1:sch 2</a>(<a class="txt" href="#E4:47_1_2"><span class="lab"><font color="Green" title="E41">A16</font></span></a>, <a class="txt" href="#E2:47_1_2"><span class="lab"><font color="Green" title="E40">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="E7:47_1_2"/><span class="lab"><font color="Green" title="E42">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="E8:47_1_2"/><span class="lab"><font color="Green" title="E43">A18</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">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="E9:47_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="#E1:47"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E8:47_1_2"><span class="lab"><font color="Green" title="E43">A18</font></span></a>, <a class="ref" href="hallmar1.html#D7" target="_self" onmouseover="rs('hallmar1/D7')" onmouseout="rh()">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E10:47_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="#E7:47_1_2"><span class="lab"><font color="Green" title="E42">A17</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>
