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

<b>set </b><font color="Maroon">S</font> = <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">P</font> =  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> ;<br/>
<a NAME="E1:42_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span>  <a href="orders_1.html#R2">partially_orders</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:42_1_1_1"/>
 <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span>  <a href="relat_2.html#R1">is_reflexive_in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 <b>by </b><i><a class="ref" href="wellord2.html#T29">WELLORD2:29</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D7">ORDERS_1:def 7</a><br/>

<b>thus </b><a NAME="E2:42_1_1_1"/>
 <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span>  <a href="relat_2.html#R8">is_transitive_in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 <b>by </b><i><a class="ref" href="wellord2.html#T30">WELLORD2:30</a></i>;<br/>

<b>thus </b><a NAME="E3:42_1_1_1"/>
 <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span>  <a href="relat_2.html#R4">is_antisymmetric_in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 <b>by </b><i><a class="ref" href="wellord2.html#T31">WELLORD2:31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:42_1_1"/><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 
<b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E3:42_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span>  <a href="orders_1.html#R4">has_upper_Zorn_property_wrt</a>  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D9">ORDERS_1:def 9</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:42_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 <b>and </b><br/><a NAME="E2:42_1_1_2"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> </span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font> <a href="orders_1.html#NR3" target="_self">is_linear-order</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_1_2_1"/>
( <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:42_1_1_2_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>take </b><font color="Maroon">x</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7">-Ideal</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_2_1_1_1"/>
not <font color="Maroon">x</font> is <a href="ideal_1.html#V5">proper</a>
 ;<br/><a NAME="E2:42_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#D5">IDEAL_1:def 5</a></i>;<br/><a NAME="E3:42_1_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="ideal_1.html#T47">IDEAL_1:47</a></i>;<br/><a NAME="E4:42_1_1_2_1_1_1"/><b>then </b>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="ring_1.html#T3" target="_self">Th3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:42_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:42_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 ;<br/><b>thus </b><a NAME="E4:42_1_1_2_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span>  )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:42_1_1_2_1_2"/>
not <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:42_1_1_2_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:42_1_1_2_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">A</font> being   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E6:42_1_1_2_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E7:42_1_1_2_1_2"/>
<font color="Maroon">A</font> is <a href="ideal_1.html#V5">proper</a>
 ;<br/><b>consider </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:42_1_1_2_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>take </b><font color="Maroon">x</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">Y</font>;<br/><a NAME="E10:42_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:42_1_1_2_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:42_1_1_2_1_2_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E4:42_1_1_2_1_2_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:42_1_1_2_1_2_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="ring_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E6:42_1_1_2_1_2_1"/><b>then </b>
 ex <font color="Olive">A</font> being  <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> st <br/>( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> )
 
;<br/>
<b>hence </b><a NAME="E7:42_1_1_2_1_2_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">R</font>
 <b>by </b><i><a class="ref" href="ring_1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> ;<br/><a NAME="E12:42_1_1_2_1_2"/><i><font color="Green">E72</font></i>: 
not <font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="ref" href="ring_1.html#T3" target="_self">Th3</a>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E13:42_1_1_2_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">B</font> is <a href="ideal_1.html#V1">add-closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D1">IDEAL_1:def 1</a><br/>

<b>assume </b><a NAME="E1:42_1_1_2_1_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Aa</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:42_1_1_2_1_2_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Aa</font>
 <b>and </b><br/><a NAME="E4:42_1_1_2_1_2_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">Aa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>assume </b><a NAME="E5:42_1_1_2_1_2_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Ab</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:42_1_1_2_1_2_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ab</font>
 <b>and </b><br/><a NAME="E8:42_1_1_2_1_2_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">Ab</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<div><i><font color="Green">E82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_1_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font>;<br/><b>assume </b><a NAME="E1:42_1_1_2_1_2_2_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><a NAME="E2:42_1_1_2_1_2_2_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#D1">IDEAL_1:def 1</a></i>;<br/><b>assume </b><a NAME="E3:42_1_1_2_1_2_2_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><b>hence </b><a NAME="E4:42_1_1_2_1_2_2_1"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:42_1_1_2_1_2_2"/>
<font color="Maroon">Aa</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">Ia</font> being   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E12:42_1_1_2_1_2_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">Aa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ia</font>
 <b>and </b><br/><a NAME="E13:42_1_1_2_1_2_2"/>
<font color="Maroon">Ia</font> is <a href="ideal_1.html#V5">proper</a>
 ;<br/>
<a NAME="E14:42_1_1_2_1_2_2"/>
<font color="Maroon">Ab</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">Ib</font> being   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E16:42_1_1_2_1_2_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">Ab</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ib</font>
 <b>and </b><br/><a NAME="E17:42_1_1_2_1_2_2"/>
<font color="Maroon">Ib</font> is <a href="ideal_1.html#V5">proper</a>
 ;<br/>
<a NAME="E18:42_1_1_2_1_2_2"/>
<span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> </span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font> is <a href="relat_2.html#V6">connected</a>
 
<b>by </b><i><a class="ref" href="ring_1.html#T2" target="_self">Th2</a>, <a class="ref" href="orders_1.html#D5">ORDERS_1:def 5</a></i>;<br/>
<a NAME="E19:42_1_1_2_1_2_2"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> </span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p3">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> </span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_2.html#D14">RELAT_2:def 14</a></i>;<br/>
<a NAME="E20:42_1_1_2_1_2_2"/>
 <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span>  is <a href="relat_2.html#V1">reflexive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_1_2_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:42_1_1_2_1_2_2_2"/>
 <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span>  <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> </span>)</span>
 <b>by </b><i>, <a class="ref" href="wellord2.html#T29">WELLORD2:29</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a><br/>


</div><b>end;</b></div>
<a NAME="E21:42_1_1_2_1_2_2"/><b>then </b>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p3">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> </span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, , <a class="ref" href="orders_1.html#T181">ORDERS_1:181</a></i>;<br/>
<a NAME="E22:42_1_1_2_1_2_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Aa</font>,<font color="Maroon">Ab</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> </span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Ab</font>,<font color="Maroon">Aa</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> </span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Y</font> or <font color="Maroon">Aa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ab</font> )
 
<b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1"><i><font color="Green">E42</font></i></a>, , , <a class="ref" href="relat_2.html#D6">RELAT_2:def 6</a></i>;<br/>
<a NAME="E23:42_1_1_2_1_2_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Aa</font>,<font color="Maroon">Ab</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span>  or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Ab</font>,<font color="Maroon">Aa</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span>  or <font color="Maroon">Aa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ab</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E24:42_1_1_2_1_2_2"/><b>then </b>
( <font color="Maroon">Aa</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ab</font> or <font color="Maroon">Ab</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Aa</font> )
 
<b>by </b><i>, , , <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E25:42_1_1_2_1_2_2"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="ring_1.html#D4" target="_self">Def4</a>, , , , , <a class="ref" href="ring_1.html#D3" target="_self">Def3</a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E14:42_1_1_2_1_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">B</font> is <a href="ideal_1.html#V2">left-ideal</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D2">IDEAL_1:def 2</a><br/>

<b>assume </b><a NAME="E1:42_1_1_2_1_2_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Aa</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:42_1_1_2_1_2_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Aa</font>
 <b>and </b><br/><a NAME="E4:42_1_1_2_1_2_3"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">Aa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:42_1_1_2_1_2_3"/>
<font color="Maroon">Aa</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="ring_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>then consider </b><font color="Maroon">Ia</font> being   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E7:42_1_1_2_1_2_3"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">Aa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ia</font>
 <b>and </b><br/><a NAME="E8:42_1_1_2_1_2_3"/>
<font color="Maroon">Ia</font> is <a href="ideal_1.html#V5">proper</a>
 ;<br/>
<a NAME="E9:42_1_1_2_1_2_3"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">p</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ia</font>
 
<b>by </b><i>, , <a class="ref" href="ideal_1.html#D2">IDEAL_1:def 2</a></i>;<br/>
<a NAME="E10:42_1_1_2_1_2_3"/>
<font color="Maroon">Ia</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="ring_1.html#D3" target="_self">Def3</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E11:42_1_1_2_1_2_3"/>
<font color="Maroon">p</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="ring_1.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:42_1_1_2_1_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">B</font> is <a href="ideal_1.html#V3">right-ideal</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D3">IDEAL_1:def 3</a><br/>

<b>assume </b><a NAME="E1:42_1_1_2_1_2_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Aa</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:42_1_1_2_1_2_4"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Aa</font>
 <b>and </b><br/><a NAME="E4:42_1_1_2_1_2_4"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">Aa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:42_1_1_2_1_2_4"/>
<font color="Maroon">Aa</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">Ia</font> being   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E7:42_1_1_2_1_2_4"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">Aa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ia</font>
 <b>and </b><br/><a NAME="E8:42_1_1_2_1_2_4"/>
<font color="Maroon">Ia</font> is <a href="ideal_1.html#V5">proper</a>
 ;<br/>
<a NAME="E9:42_1_1_2_1_2_4"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ia</font>
 
<b>by </b><i><a class="ref" href="ring_1.html#D3" target="_self">Def3</a>, <a class="ref" href="ring_1.html#D4" target="_self">Def4</a>, <a class="ref" href="ideal_1.html#D3">IDEAL_1:def 3</a></i>;<br/>
<a NAME="E10:42_1_1_2_1_2_4"/>
<font color="Maroon">Ia</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="ring_1.html#D4" target="_self">Def4</a>, , <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E11:42_1_1_2_1_2_4"/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_1_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_2_1_2_5"/>
not <font color="Maroon">B</font> is <a href="ideal_1.html#V5">proper</a>
 ;<br/><a NAME="E2:42_1_1_2_1_2_5"/><b>then </b>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="ring_1.html#D1" target="_self">Def1</a>, , <a class="ref" href="ideal_1.html#T19">IDEAL_1:19</a></i>;<br/><b>then consider </b><font color="Maroon">Aa</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:42_1_1_2_1_2_5"/><i><font color="Green">E99</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Aa</font>
 <b>and </b><br/><a NAME="E5:42_1_1_2_1_2_5"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">Aa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:42_1_1_2_1_2_5"/>
<font color="Maroon">Aa</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 <b>by </b><i>, </i>;<br/><a NAME="E7:42_1_1_2_1_2_5"/><b>then </b>
 ex <font color="Olive">Ia</font> being  <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> st <br/>( <font color="Maroon">Aa</font> <a href="hidden.html#R1">=</a> <font color="Olive">Ia</font> &amp; <font color="Olive">Ia</font> is <a href="ideal_1.html#V5">proper</a> )
 ;<br/><b>hence </b><a NAME="E8:42_1_1_2_1_2_5"/>
contradiction
 <b>by </b><i><a class="ref" href="ring_1.html#D3" target="_self">Def3</a>, <a class="ref" href="ideal_1.html#T19">IDEAL_1:19</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E17:42_1_1_2_1_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="ring_1.html#D1" target="_self">Def1</a>, <a class="ref" href="ring_1.html#D2" target="_self">Def2</a>, </i>;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E18:42_1_1_2_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E19:42_1_1_2_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E20:42_1_1_2_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 <b>by </b><i>, , <a class="ref" href="ring_1.html#D3" target="_self">Def3</a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:42_1_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">x</font> <a href="orders_1.html#R6">is_maximal_in</a>  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 <b>by </b><i>, , <a class="ref" href="orders_1.html#T173">ORDERS_1:173</a></i>;<br/>
<a NAME="E6:42_1_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="orders_1.html#D11">ORDERS_1:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon">I</font> being   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E8:42_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>and </b><br/><a NAME="E9:42_1_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">I</font> is <a href="ideal_1.html#V5">proper</a>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">I</font>
;<br/>

<b>thus </b><a NAME="E10:42_1_1"/>
<font color="Maroon">I</font> is <a href="ideal_1.html#V5">proper</a>
 <b>by </b><i><a class="ref" href="ring_1.html#T3" target="_self">Th3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ring_1.html#D4" target="_self">RING_1:def 4</a><br/>

<b>let </b><font color="Maroon">J</font> be   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ring_1.html#D3" target="_self">RING_1:def 3</a><br/>

<b>assume </b><a NAME="E11:42_1_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">J</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_3"/>
<font color="Maroon">J</font> is <a href="ideal_1.html#V5">proper</a>
 ;<br/><a NAME="E2:42_1_1_3"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">J</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 ;<br/><a NAME="E3:42_1_1_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">J</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> : <font color="Olive">A</font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="ring_1.html#T2" target="_self">Th2</a>, , , <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/><b>hence </b><a NAME="E4:42_1_1_3"/>
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font>
 <b>by </b><i>, <a class="ref" href="ring_1.html#D1" target="_self">Def1</a>, , , <a class="ref" href="orders_1.html#D11">ORDERS_1:def 11</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:42_1_1"/>
( <font color="Maroon">J</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> or  not <font color="Maroon">J</font> is <a href="ideal_1.html#V5">proper</a> )
 ;<br/>


</div>
