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

<b>let </b><font color="Maroon">X</font> be    <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:56"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1"/>
( <font color="Maroon">X</font> is <a href="realset2.html#V3">trivial</a> or  not <font color="Maroon">X</font> is <a href="realset2.html#V3">trivial</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1"><b>suppose </b></a><a NAME="E1:56_1_1"/>
<font color="Maroon">X</font> is <a href="realset2.html#V3">trivial</a>
 ;<br/><div class="add"><a NAME="E2:56_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i><a class="ref" href="realset2.html#D5">REALSET2:def 5</a></i>;<br/><b>thus </b><a NAME="E3:56_1_1"/>
<font color="Maroon">X</font> is <a href="necklace.html#V1">symmetric</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_1_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> or  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E2:56_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:56_1_1_1_1_1"/><i><font color="Green">E35</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="necklace.html#D4">NECKLACE:def 4</a>,<a class="ref" href="relat_2.html#D3">RELAT_2:def 3</a><br/><b>assume </b><a NAME="E2:56_1_1_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">X</font> )
 ;<br/><b>hence </b><a NAME="E3:56_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:56_1_1_1_1_1"><i><font color="Green">E35</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:56_1_1_1_1_2"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:56_1_1_1_1_2"/><i><font color="Green">E36</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E4:56_1_1_1_1_2"/><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T35">ZFMISC_1:35</a></i>;<br/><b>thus </b><a NAME="E5:56_1_1_1_1_2"/>
<font color="Maroon">X</font> is <a href="necklace.html#V1">symmetric</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_1_1_1_2_1_1"/>
( the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E3:56_1_1_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:56_1_1_1_1_2_1_1_1"/><i><font color="Green">E41</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="necklace.html#D4">NECKLACE:def 4</a>,<a class="ref" href="relat_2.html#D3">RELAT_2:def 3</a><br/><b>assume </b><a NAME="E2:56_1_1_1_1_2_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">X</font> )
 ;<br/><b>hence </b><a NAME="E3:56_1_1_1_1_2_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">X</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:56_1_1_1_1_2_1_1_2"/><i><font color="Green">E42</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="necklace.html#D4">NECKLACE:def 4</a>,<a class="ref" href="relat_2.html#D3">RELAT_2:def 3</a><br/><b>assume </b><a NAME="E2:56_1_1_1_1_2_1_1_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">X</font> )
 ;<br/><a NAME="E3:56_1_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:56_1_1_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E5:56_1_1_1_1_2_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">X</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2"><b>suppose </b></a><a NAME="E1:56_1_2"/><i><font color="Green">E60</font></i>: 
not <font color="Maroon">X</font> is <a href="realset2.html#V3">trivial</a>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">X</font> being non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a>   st not <font color="Olive">X</font> is <a href="realset2.html#V3">trivial</a> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a>  &amp;  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">X</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">X</font> is <a href="necklace.html#V1">symmetric</a>;<br/><a NAME="E2:56_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:56_1_2_1"/><i><font color="Green">E62</font></i>: 
( not <font color="Maroon">X</font> is <a href="realset2.html#V3">trivial</a> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a>  )
 ;<br/>

<b>consider </b><font color="Maroon">R</font> being  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><br/><a NAME="E3:56_1_2_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a>  )
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D4">NECKLA_2:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">cX</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>;<br/>
<b>reconsider </b><font color="Maroon">cX</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="neckla_2.html#T3">NECKLA_2:3</a></i>;<br/>
<b>assume </b><a NAME="E5:56_1_2_1"/><i><font color="Green">E64</font></i>: 
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> 0
 ;<br/>

<b>assume </b><a NAME="E6:56_1_2_1"/>
not <font color="Maroon">X</font> is <a href="necklace.html#V1">symmetric</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_2_1_1"/>
(  <a href="card_1.html#K4">card</a> <font color="Maroon">cX</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  <a href="card_1.html#K4">card</a> <font color="Maroon">cX</font> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i>, <a class="ref" href="card_2.html#T19">CARD_2:19</a>, <a class="ref" href="card_2.html#T57">CARD_2:57</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:56_1_2_1_1_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cX</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:56_1_2_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:56_1_2_1_1_2"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cX</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:56_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="graph_5.html#T5">GRAPH_5:5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:56_1_2"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:56_1_2_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>let </b><font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E2:56_1_2_2"/><i><font color="Green">E93</font></i>: 
( not <font color="Maroon">Y</font> is <a href="realset2.html#V3">trivial</a> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a>  )
 ;<br/>

<b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon">cY</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>;<br/>
<b>consider </b><font color="Maroon">R</font> being  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><br/><a NAME="E5:56_1_2_2"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a>  )
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D4">NECKLA_2:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">cY</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="neckla_2.html#T3">NECKLA_2:3</a></i>;<br/>
<b>consider </b><font color="Maroon">H1</font> being  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">H2</font> being  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><br/><a NAME="E8:56_1_2_2"/><i><font color="Green">E104</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> &amp; <font color="Maroon">H1</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a>  &amp; <font color="Maroon">H2</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4">fin_RelStr_sp</a>  )
 <b>and </b><br/><a NAME="E9:56_1_2_2"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1">union_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font> or <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font> )
 <b>by </b><i>, <a class="ref" href="neckla_2.html#T6">NECKLA_2:6</a></i>;<br/>
<b>set </b><font color="Maroon">cH1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>;<br/>
<b>set </b><font color="Maroon">cH2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>;<br/>
<a NAME="E10:56_1_2_2"/><i><font color="Green">E106</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">cY</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="neckla_2.html#D2">NECKLA_2:def 2</a>, <a class="ref" href="neckla_2.html#D3">NECKLA_2:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">R1</font> being  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><br/><a NAME="E12:56_1_2_2"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R1</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R1</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a>  )
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D4">NECKLA_2:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">cH1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="neckla_2.html#T3">NECKLA_2:3</a></i>;<br/>
<b>consider </b><font color="Maroon">R2</font> being  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><br/><a NAME="E15:56_1_2_2"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">H2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R2</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R2</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a>  )
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D4">NECKLA_2:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">cH2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="neckla_3.html#T3" target="_self">Th3</a>, <a class="ref" href="neckla_2.html#T3">NECKLA_2:3</a></i>;<br/>
<a NAME="E17:56_1_2_2"/><i><font color="Green">E109</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">cY</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>;<br/>
<b>assume </b><a NAME="E18:56_1_2_2"/>
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>

<a NAME="E19:56_1_2_2"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">cY</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/>
<a NAME="E20:56_1_2_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cY</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T57">CARD_2:57</a></i>;<br/>
<a NAME="E21:56_1_2_2"/><b>then </b><i><font color="Green">E110</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">cY</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/>
<a NAME="E22:56_1_2_2"/><i><font color="Green">E111</font></i>: 
( not <font color="Maroon">H1</font> is <a href="struct_0.html#V3">empty</a> &amp; not <font color="Maroon">H2</font> is <a href="struct_0.html#V3">empty</a> )
 
<b>by </b><i>, <a class="ref" href="neckla_2.html#T4">NECKLA_2:4</a></i>;<br/>
<a NAME="E23:56_1_2_2"/><b>then </b>
( not <font color="Maroon">cH1</font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">cH2</font> is <a href="xboole_0.html#V1">empty</a> )
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E24:56_1_2_2"/><b>then </b>
(  <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="card_4.html#T17">CARD_4:17</a></i>;<br/>
<a NAME="E25:56_1_2_2"/><b>then </b><i><font color="Green">E112</font></i>: 
(  <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_2_2_1"/>
(  <a href="card_1.html#K4">card</a> <font color="Maroon">cY</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> or (  <a href="card_1.html#K4">card</a> <font color="Maroon">cY</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 ) or ( <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> <a href="hidden.html#R1">=</a> 1 ) or ( <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 ) or ( <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> <a href="hidden.html#R1">=</a> 1 ) or ( <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 ) )
 </span><b>by </b><i>, , , <a class="ref" href="nat_1.html#T26">NAT_1:26</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="56_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:56_1_2_2_1_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cY</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:56_1_2_2_1_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cY</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/><a NAME="E3:56_1_2_2_1_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="card_2.html#T57">CARD_2:57</a></i>;<br/><a NAME="E4:56_1_2_2_1_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/><b>hence </b><a NAME="E5:56_1_2_2_1_1"/>
<font color="Maroon">Y</font> is <a href="necklace.html#V1">symmetric</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:56_1_2_2_1_2"/><i><font color="Green">E113</font></i>: 
(  <a href="card_1.html#K4">card</a> <font color="Maroon">cY</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E3:56_1_2_2_1_2"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cY</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/><a NAME="E4:56_1_2_2_1_2"/><b>then </b>
<font color="Maroon">cY</font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:56_1_2_2_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">cY</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="card_1.html#T48">CARD_1:48</a></i>;<br/><a NAME="E7:56_1_2_2_1_2"/>
not <font color="Maroon">cY</font> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i>, <a class="ref" href="realset2.html#D5">REALSET2:def 5</a></i>;<br/><b>hence </b><a NAME="E8:56_1_2_2_1_2"/>
<font color="Maroon">Y</font> is <a href="necklace.html#V1">symmetric</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_2_1_3"><b>suppose </b></a><a NAME="E1:56_1_2_2_1_3"/><i><font color="Green">E115</font></i>: 
( <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:56_1_2_2_1_3"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">cH1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:56_1_2_2_1_3"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">cH2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/><a NAME="E6:56_1_2_2_1_3"/>
( not <font color="Maroon">cH1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">cH1</font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i/>;<br/><a NAME="E7:56_1_2_2_1_3"/><b>then </b><i><font color="Green">E118</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="relat_2.html#R3">is_symmetric_in</a> <font color="Maroon">cH1</font>
 <b>by </b><i/>;<br/><a NAME="E8:56_1_2_2_1_3"/>
( not <font color="Maroon">cH2</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">cH2</font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i/>;<br/><a NAME="E9:56_1_2_2_1_3"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="relat_2.html#R3">is_symmetric_in</a> <font color="Maroon">cH2</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">H2</font> = <font color="Maroon">H2</font> as  <a href="necklace.html#V1">symmetric</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i><a class="ref" href="necklace.html#D4">NECKLACE:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">H1</font> = <font color="Maroon">H1</font> as  <a href="necklace.html#V1">symmetric</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i>, <a class="ref" href="necklace.html#D4">NECKLACE:def 4</a></i>;<br/><a NAME="E12:56_1_2_2_1_3"/>
(  <a href="neckla_2.html#K1">union_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font> is <a href="necklace.html#V1">symmetric</a> &amp;  <a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font> is <a href="necklace.html#V1">symmetric</a> )
 ;<br/><b>hence </b><a NAME="E13:56_1_2_2_1_3"/>
<font color="Maroon">Y</font> is <a href="necklace.html#V1">symmetric</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_2_1_4"><b>suppose </b></a><a NAME="E1:56_1_2_2_1_4"/><i><font color="Green">E119</font></i>: 
( <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:56_1_2_2_1_4"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">cH1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/><a NAME="E4:56_1_2_2_1_4"/>
<font color="Maroon">cH1</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="realset1.html#V1">trivial</a>  <a href="hidden.html#M1">set</a> 
 <b>by </b><i/>;<br/><a NAME="E5:56_1_2_2_1_4"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H1</font> <a href="relat_2.html#R3">is_symmetric_in</a> <font color="Maroon">cH1</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">H1</font> = <font color="Maroon">H1</font> as  <a href="necklace.html#V1">symmetric</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i><a class="ref" href="necklace.html#D4">NECKLACE:def 4</a></i>;<br/><a NAME="E7:56_1_2_2_1_4"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:56_1_2_2_1_4"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:56_1_2_2_1_4"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/><a NAME="E10:56_1_2_2_1_4"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">cH2</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="card_2.html#T57">CARD_2:57</a></i>;<br/><a NAME="E11:56_1_2_2_1_4"/><b>then </b><i><font color="Green">E121</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">cH2</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/><a NAME="E12:56_1_2_2_1_4"/>
not  <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i>, <a class="ref" href="nat_2.html#T30">NAT_2:30</a></i>;<br/><a NAME="E13:56_1_2_2_1_4"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 <b>by </b><i><a class="ref" href="nat_2.html#T31">NAT_2:31</a></i>;<br/><a NAME="E14:56_1_2_2_1_4"/><b>then </b>
( not <font color="Maroon">cH2</font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">cH2</font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i>, <a class="ref" href="realset1.html#T13">REALSET1:13</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E15:56_1_2_2_1_4"/><b>then </b>
( not <font color="Maroon">H2</font> is <a href="struct_0.html#V3">empty</a> &amp; not <font color="Maroon">H2</font> is <a href="realset2.html#V3">trivial</a> )
 <b>by </b><i><a class="ref" href="realset2.html#D5">REALSET2:def 5</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">H2</font> = <font color="Maroon">H2</font> as  <a href="necklace.html#V1">symmetric</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i>, , </i>;<br/><a NAME="E17:56_1_2_2_1_4"/>
(  <a href="neckla_2.html#K1">union_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font> is <a href="necklace.html#V1">symmetric</a> &amp;  <a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font> is <a href="necklace.html#V1">symmetric</a> )
 ;<br/><b>hence </b><a NAME="E18:56_1_2_2_1_4"/>
<font color="Maroon">Y</font> is <a href="necklace.html#V1">symmetric</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_2_1_5"><b>suppose </b></a><a NAME="E1:56_1_2_2_1_5"/><i><font color="Green">E122</font></i>: 
( <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:56_1_2_2_1_5"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">cH2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/><a NAME="E4:56_1_2_2_1_5"/>
<font color="Maroon">cH2</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="realset1.html#V1">trivial</a>  <a href="hidden.html#M1">set</a> 
 <b>by </b><i/>;<br/><a NAME="E5:56_1_2_2_1_5"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">H2</font> <a href="relat_2.html#R3">is_symmetric_in</a> <font color="Maroon">cH2</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">H2</font> = <font color="Maroon">H2</font> as  <a href="necklace.html#V1">symmetric</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i><a class="ref" href="necklace.html#D4">NECKLACE:def 4</a></i>;<br/><a NAME="E7:56_1_2_2_1_5"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:56_1_2_2_1_5"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:56_1_2_2_1_5"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/><a NAME="E10:56_1_2_2_1_5"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">cH1</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="card_2.html#T57">CARD_2:57</a></i>;<br/><a NAME="E11:56_1_2_2_1_5"/><b>then </b><i><font color="Green">E124</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">cH1</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/><a NAME="E12:56_1_2_2_1_5"/>
not  <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i>, <a class="ref" href="nat_2.html#T30">NAT_2:30</a></i>;<br/><a NAME="E13:56_1_2_2_1_5"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 <b>by </b><i><a class="ref" href="nat_2.html#T31">NAT_2:31</a></i>;<br/><a NAME="E14:56_1_2_2_1_5"/><b>then </b>
( not <font color="Maroon">cH1</font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">cH1</font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i>, <a class="ref" href="realset1.html#T13">REALSET1:13</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E15:56_1_2_2_1_5"/><b>then </b>
( not <font color="Maroon">H1</font> is <a href="struct_0.html#V3">empty</a> &amp; not <font color="Maroon">H1</font> is <a href="realset2.html#V3">trivial</a> )
 <b>by </b><i><a class="ref" href="realset2.html#D5">REALSET2:def 5</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">H1</font> = <font color="Maroon">H1</font> as  <a href="necklace.html#V1">symmetric</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i>, , </i>;<br/><a NAME="E17:56_1_2_2_1_5"/>
(  <a href="neckla_2.html#K1">union_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font> is <a href="necklace.html#V1">symmetric</a> &amp;  <a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font> is <a href="necklace.html#V1">symmetric</a> )
 ;<br/><b>hence </b><a NAME="E18:56_1_2_2_1_5"/>
<font color="Maroon">Y</font> is <a href="necklace.html#V1">symmetric</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_2_1_6"><b>suppose </b></a><a NAME="E1:56_1_2_2_1_6"/><i><font color="Green">E125</font></i>: 
( <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 ;<br/><div class="add"><a NAME="E2:56_1_2_2_1_6"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2_2_1_6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:56_1_2_2_1_6_1"/>
not  <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>

<a NAME="E2:56_1_2_2_1_6_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E3:56_1_2_2_1_6_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:56_1_2_2_1_6"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:56_1_2_2_1_6"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/><a NAME="E5:56_1_2_2_1_6"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">cH1</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="card_2.html#T57">CARD_2:57</a></i>;<br/><a NAME="E6:56_1_2_2_1_6"/><b>then </b><i><font color="Green">E126</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">cH1</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/><a NAME="E7:56_1_2_2_1_6"/>
not  <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i>, <a class="ref" href="nat_2.html#T30">NAT_2:30</a></i>;<br/><a NAME="E8:56_1_2_2_1_6"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 <b>by </b><i><a class="ref" href="nat_2.html#T31">NAT_2:31</a></i>;<br/><a NAME="E9:56_1_2_2_1_6"/><b>then </b>
( not <font color="Maroon">cH1</font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">cH1</font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i>, <a class="ref" href="realset1.html#T13">REALSET1:13</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E10:56_1_2_2_1_6"/><b>then </b>
( not <font color="Maroon">H1</font> is <a href="struct_0.html#V3">empty</a> &amp; not <font color="Maroon">H1</font> is <a href="realset2.html#V3">trivial</a> )
 <b>by </b><i><a class="ref" href="realset2.html#D5">REALSET2:def 5</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">H1</font> = <font color="Maroon">H1</font> as  <a href="necklace.html#V1">symmetric</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i>, , </i>;<br/><a NAME="E12:56_1_2_2_1_6"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2_2_1_6_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:56_1_2_2_1_6_2"/>
not  <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>

<a NAME="E2:56_1_2_2_1_6_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E3:56_1_2_2_1_6_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:56_1_2_2_1_6"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E14:56_1_2_2_1_6"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/><a NAME="E15:56_1_2_2_1_6"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">cH2</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="card_2.html#T57">CARD_2:57</a></i>;<br/><a NAME="E16:56_1_2_2_1_6"/><b>then </b><i><font color="Green">E127</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">cH2</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/><a NAME="E17:56_1_2_2_1_6"/>
not  <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i>, <a class="ref" href="nat_2.html#T30">NAT_2:30</a></i>;<br/><a NAME="E18:56_1_2_2_1_6"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">cH2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 <b>by </b><i><a class="ref" href="nat_2.html#T31">NAT_2:31</a></i>;<br/><a NAME="E19:56_1_2_2_1_6"/><b>then </b>
( not <font color="Maroon">cH2</font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">cH2</font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i>, <a class="ref" href="realset1.html#T13">REALSET1:13</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E20:56_1_2_2_1_6"/><b>then </b>
( not <font color="Maroon">H2</font> is <a href="struct_0.html#V3">empty</a> &amp; not <font color="Maroon">H2</font> is <a href="realset2.html#V3">trivial</a> )
 <b>by </b><i><a class="ref" href="realset2.html#D5">REALSET2:def 5</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">H2</font> = <font color="Maroon">H2</font> as  <a href="necklace.html#V1">symmetric</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i>, , </i>;<br/><a NAME="E22:56_1_2_2_1_6"/>
(  <a href="neckla_2.html#K1">union_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font> is <a href="necklace.html#V1">symmetric</a> &amp;  <a href="neckla_2.html#K2">sum_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font> is <a href="necklace.html#V1">symmetric</a> )
 ;<br/><b>hence </b><a NAME="E23:56_1_2_2_1_6"/>
<font color="Maroon">Y</font> is <a href="necklace.html#V1">symmetric</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:56_1_2"/><i><font color="Green">E128</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i><b>consider </b><font color="Maroon">R</font> being  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><br/><a NAME="E6:56_1_2"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a>  )
 <b>by </b><i>, <a class="ref" href="neckla_2.html#D4">NECKLA_2:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i>, <a class="ref" href="neckla_2.html#T4">NECKLA_2:4</a></i>;<br/><a NAME="E8:56_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="neckla_2.html#T3">NECKLA_2:3</a></i>;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:56_1_2"/><i><font color="Green">E130</font></i>: 
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="card_4.html#T4">CARD_4:4</a></i>;<br/><b>thus </b><a NAME="E11:56_1_2"/>
<font color="Maroon">X</font> is <a href="necklace.html#V1">symmetric</a>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
