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

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> ;<br/>
<a NAME="E1:42_1_1"/><i><font color="Green">E26</font></i>: 
 <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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">E27</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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></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">E28</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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">E29</font></i>: 
<span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> </span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">c<sub>4</sub></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">E30</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></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">c<sub>5</sub></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">E31</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>5</sub></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">c<sub>1</sub></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">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="ring_1.html#E2:42_1_1_2_1_1_1"><i><font color="Green">E31</font></i></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">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></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">c<sub>5</sub></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">b<sub>2</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>2</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span>  )
 <b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1_2_1_1"><i><font color="Green">E30</font></i></a></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">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>5</sub></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">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="ring_1.html#E3:42_1_1_2_1_2"><i><font color="Green">E30</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:42_1_1_2_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E7:42_1_1_2_1_2"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="ideal_1.html#V5">proper</a>
 ;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></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">E32</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></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">c<sub>8</sub></font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>4</sub></font>;<br/><a NAME="E10:42_1_1_2_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>9</sub></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">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></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">E33</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E4:42_1_1_2_1_2_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="ring_1.html#E4:42_1_1_2_1_2_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E6:42_1_1_2_1_2_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="ring_1.html#E3:42_1_1_2_1_2_1"><i><font color="Green">E33</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E12:42_1_1_2_1_2"/><i><font color="Green">E33</font></i>: 
not <font color="Maroon">c<sub>9</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="ring_1.html#E9:42_1_1_2_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="ring_1.html#E6:42_1_1_2_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="ring_1.html#E3:42_1_1_2_1_2"><i><font color="Green">E30</font></i></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">E34</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></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">E35</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E4:42_1_1_2_1_2_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></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">E37</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E8:42_1_1_2_1_2_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<div><i><font color="Green">E39</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">c<sub>14</sub></font> be   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:42_1_1_2_1_2_2_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> )
 ;<br/><a NAME="E2:42_1_1_2_1_2_2_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>hence </b><a NAME="E4:42_1_1_2_1_2_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="ring_1.html#E2:42_1_1_2_1_2_2_1"><i><font color="Green">E40</font></i></a>, <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">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="ring_1.html#E4:42_1_1_2_1_2_2"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E12:42_1_1_2_1_2_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E13:42_1_1_2_1_2_2"/>
<font color="Maroon">c<sub>14</sub></font> is <a href="ideal_1.html#V5">proper</a>
 ;<br/>
<a NAME="E14:42_1_1_2_1_2_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="ring_1.html#E8:42_1_1_2_1_2_2"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E16:42_1_1_2_1_2_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E17:42_1_1_2_1_2_2"/>
<font color="Maroon">c<sub>15</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> </span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>4</sub></font> is <a href="relat_2.html#V6">connected</a>
 
<b>by </b><i><a class="txt" href="ring_1.html#E2:42_1_1_2"><i><font color="Green">E29</font></i></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">E42</font></i>: 
<span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> </span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>4</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> </span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>4</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> </span>)</span>
 <b>by </b><i><a class="txt" href="ring_1.html#E2:42_1_1"><i><font color="Green">E27</font></i></a>, <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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> </span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="ring_1.html#E2:42_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="ring_1.html#E1:42_1_1_2"><i><font color="Green">E28</font></i></a>, <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">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> </span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>4</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> </span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="ring_1.html#E19:42_1_1_2_1_2_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="ring_1.html#E4:42_1_1_2_1_2_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="ring_1.html#E8:42_1_1_2_1_2_2"><i><font color="Green">E38</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">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span>  or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></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">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="ring_1.html#E8:42_1_1_2_1_2_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="ring_1.html#E4:42_1_1_2_1_2_2"><i><font color="Green">E36</font></i></a>, <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">c<sub>10</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="ring_1.html#E9:42_1_1_2_1_2_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="ring_1.html#E12:42_1_1_2_1_2_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="ring_1.html#E16:42_1_1_2_1_2_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="ring_1.html#E3:42_1_1_2_1_2_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="ring_1.html#E4:42_1_1_2_1_2_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="ring_1.html#E7:42_1_1_2_1_2_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="ring_1.html#E8:42_1_1_2_1_2_2"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:42_1_1_2_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></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">E36</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E4:42_1_1_2_1_2_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="ring_1.html#E4:42_1_1_2_1_2_3"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:42_1_1_2_1_2_3"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E8:42_1_1_2_1_2_3"/>
<font color="Maroon">c<sub>13</sub></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">E39</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="ring_1.html#E3:42_1_1_2_1_2_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="ring_1.html#E7:42_1_1_2_1_2_3"><i><font color="Green">E38</font></i></a>, <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">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="ring_1.html#E7:42_1_1_2_1_2_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="ring_1.html#E4:42_1_1_2_1_2_3"><i><font color="Green">E37</font></i></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">c<sub>10</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="ring_1.html#E9:42_1_1_2_1_2_3"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:42_1_1_2_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></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">E37</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E4:42_1_1_2_1_2_4"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="ring_1.html#E4:42_1_1_2_1_2_4"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:42_1_1_2_1_2_4"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E8:42_1_1_2_1_2_4"/>
<font color="Maroon">c<sub>13</sub></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">E40</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="ring_1.html#E3:42_1_1_2_1_2_4"><i><font color="Green">E37</font></i></a>, <a class="txt" href="ring_1.html#E7:42_1_1_2_1_2_4"><i><font color="Green">E39</font></i></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">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="ring_1.html#E7:42_1_1_2_1_2_4"><i><font color="Green">E39</font></i></a>, <a class="txt" href="ring_1.html#E4:42_1_1_2_1_2_4"><i><font color="Green">E38</font></i></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">c<sub>11</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="ring_1.html#E9:42_1_1_2_1_2_4"><i><font color="Green">E40</font></i></a></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">c<sub>9</sub></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">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="ring_1.html#E12:42_1_1_2_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="ring_1.html#E14:42_1_1_2_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="ideal_1.html#T19">IDEAL_1:19</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></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">E37</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E5:42_1_1_2_1_2_5"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></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">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="ring_1.html#E5:42_1_1_2_1_2_5"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E7:42_1_1_2_1_2_5"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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="txt" href="ring_1.html#E4:42_1_1_2_1_2_5"><i><font color="Green">E37</font></i></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">E37</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="ring_1.html#E15:42_1_1_2_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="ring_1.html#E12:42_1_1_2_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="ring_1.html#E13:42_1_1_2_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="ring_1.html#E14:42_1_1_2_1_2"><i><font color="Green">E35</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></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">E38</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E19:42_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></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">c<sub>10</sub></font>,<font color="Maroon">c<sub>8</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="ring_1.html#E18:42_1_1_2_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="ring_1.html#E17:42_1_1_2_1_2"><i><font color="Green">E37</font></i></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">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:42_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>4</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="ring_1.html#E2:42_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="orders_1.html#T173">ORDERS_1:173</a></i>;<br/>
<a NAME="E6:42_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> </span>)</span>
 
<b>by </b><i><a class="txt" href="ring_1.html#E5:42_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="orders_1.html#D11">ORDERS_1:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:42_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E9:42_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="ideal_1.html#V5">proper</a>
 <b>by </b><i><a class="txt" href="ring_1.html#E2:42_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E10:42_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="ideal_1.html#V5">proper</a>
 <b>by </b><i><a class="txt" href="ring_1.html#E9:42_1_1"><i><font color="Green">E31</font></i></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">c<sub>6</sub></font> be   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></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">E32</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></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">E33</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></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">b<sub>1</sub></font> where B is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ideal_1.html#V5">proper</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="ring_1.html#E2:42_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="ring_1.html#E6:42_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="ring_1.html#E8:42_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="ring_1.html#E11:42_1_1"><i><font color="Green">E32</font></i></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">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="ring_1.html#E2:42_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="ring_1.html#E2:42_1_1_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="ring_1.html#E8:42_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="ring_1.html#E5:42_1_1"><i><font color="Green">E28</font></i></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">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> or  not <font color="Maroon">c<sub>6</sub></font> is <a href="ideal_1.html#V5">proper</a> )
 ;<br/>


</div>
