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

<b>set </b><font color="Maroon" title="c2">S</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c3">P</font> =  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> ;<br/>
<a NAME="E1:42_1_1"/><i><font color="Green" title="E26">A1</font></i>: 
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span>  <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">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" title="WELLORD2:func.1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span>  <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> 
 <b>by </b><i><a class="ref" href="wellord2.html#T29" title="WELLORD2:th.29">WELLORD2:29</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D7" title="ORDERS_1:def.7">ORDERS_1:def 7</a><br/>

<b>thus </b><a NAME="E2:42_1_1_1"/>
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span>  <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> 
 <b>by </b><i><a class="ref" href="wellord2.html#T30" title="WELLORD2:th.30">WELLORD2:30</a></i>;<br/>

<b>thus </b><a NAME="E3:42_1_1_1"/>
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span>  <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> 
 <b>by </b><i><a class="ref" href="wellord2.html#T31" title="WELLORD2:th.31">WELLORD2:31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:42_1_1"/><i><font color="Green" title="E27">A2</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> 
 
<b>by </b><i><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E3:42_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span>  <a href="orders_1.html#R4" title="ORDERS_1:pred.4">has_upper_Zorn_property_wrt</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">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" title="c4">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D9" title="ORDERS_1:def.9">ORDERS_1:def 9</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:42_1_1_2"/><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> 
 <b>and </b><br/><a NAME="E2:42_1_1_2"/><i><font color="Green" title="E29">A4</font></i>: 
<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> </span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c4">Y</font> <a href="orders_1.html#NR3" title="ORDERS_1:NR.3">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" title="c4">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c4">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">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" title="E30">A5</font></i>: 
<font color="Maroon" title="c4">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c5">x</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-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" title="c5">x</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a>
 ;<br/><a NAME="E2:42_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E31">A6</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#D5" title="IDEAL_1:def.5">IDEAL_1:def 5</a></i>;<br/><a NAME="E3:42_1_1_2_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="ideal_1.html#T47" title="IDEAL_1:th.47">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" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="ring_1.html#E2:42_1_1_2_1_1_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">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" title="VECTSP_1:def.21">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" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">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" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">A</font> where <font color="Olive" title="b2">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b2">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">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" title="E30">A5</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" title="c4">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:42_1_1_2_1_2"/><i><font color="Green" title="E30">A7</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:42_1_1_2_1_2"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1_2"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="ring_1.html#E3:42_1_1_2_1_2"><i><font color="Green" title="E30">A7</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">A</font> being   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E6:42_1_1_2_1_2"/><i><font color="Green" title="E31">A8</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font>
 <b>and </b><br/><a NAME="E7:42_1_1_2_1_2"/>
<font color="Maroon" title="c6">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a>
 ;<br/><b>consider </b><font color="Maroon" title="c7">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:42_1_1_2_1_2"/><i><font color="Green" title="E32">A9</font></i>: 
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>take </b><font color="Maroon" title="c8">x</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">Y</font>;<br/><a NAME="E10:42_1_1_2_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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" title="c9">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:42_1_1_2_1_2_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">x</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:42_1_1_2_1_2_1"/><i><font color="Green" title="E33">A10</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font>
 <b>and </b><br/><a NAME="E4:42_1_1_2_1_2_1"/><i><font color="Green" title="E34">A11</font></i>: 
<font color="Maroon" title="c10">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:42_1_1_2_1_2_1"/>
<font color="Maroon" title="c10">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1_2"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="ring_1.html#E4:42_1_1_2_1_2_1"><i><font color="Green" title="E34">A11</font></i></a></i>;<br/>
<a NAME="E6:42_1_1_2_1_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">A</font> being  <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Maroon" title="c10">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> )
 
;<br/>
<b>hence </b><a NAME="E7:42_1_1_2_1_2_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="ring_1.html#E3:42_1_1_2_1_2_1"><i><font color="Green" title="E33">A10</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c9">B</font> = <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> ;<br/><a NAME="E12:42_1_1_2_1_2"/><i><font color="Green" title="E33">A12</font></i>: 
not <font color="Maroon" title="c9">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="ring_1.html#E9:42_1_1_2_1_2"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="ring_1.html#E6:42_1_1_2_1_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="ring_1.html#E3:42_1_1_2_1_2"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E13:42_1_1_2_1_2"/><i><font color="Green" title="E34">A13</font></i>: 
<font color="Maroon" title="c9">B</font> is <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">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" title="c10">a</font>, <font color="Maroon" title="c11">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D1" title="IDEAL_1:def.1">IDEAL_1:def 1</a><br/>

<b>assume </b><a NAME="E1:42_1_1_2_1_2_2"/>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">Aa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:42_1_1_2_1_2_2"/><i><font color="Green" title="E35">A14</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Aa</font>
 <b>and </b><br/><a NAME="E4:42_1_1_2_1_2_2"/><i><font color="Green" title="E36">A15</font></i>: 
<font color="Maroon" title="c12">Aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>assume </b><a NAME="E5:42_1_1_2_1_2_2"/>
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">Ab</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:42_1_1_2_1_2_2"/><i><font color="Green" title="E37">A16</font></i>: 
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Ab</font>
 <b>and </b><br/><a NAME="E8:42_1_1_2_1_2_2"/><i><font color="Green" title="E38">A17</font></i>: 
<font color="Maroon" title="c13">Ab</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<div><i><font color="Green" title="E39">A18</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" title="c14">A</font> be   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font>;<br/><b>assume </b><a NAME="E1:42_1_1_2_1_2_2_1"/>
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> &amp; <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font> )
 ;<br/><a NAME="E2:42_1_1_2_1_2_2_1"/><b>then </b><i><font color="Green" title="E40">A19</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#D1" title="IDEAL_1:def.1">IDEAL_1:def 1</a></i>;<br/><b>assume </b><a NAME="E3:42_1_1_2_1_2_2_1"/>
<font color="Maroon" title="c14">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 ;<br/><b>hence </b><a NAME="E4:42_1_1_2_1_2_2_1"/>
<font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</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" title="E40">A19</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:42_1_1_2_1_2_2"/>
<font color="Maroon" title="c12">Aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1_2"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="ring_1.html#E4:42_1_1_2_1_2_2"><i><font color="Green" title="E36">A15</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">Ia</font> being   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E12:42_1_1_2_1_2_2"/><i><font color="Green" title="E40">A20</font></i>: 
<font color="Maroon" title="c12">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Ia</font>
 <b>and </b><br/><a NAME="E13:42_1_1_2_1_2_2"/>
<font color="Maroon" title="c14">Ia</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a>
 ;<br/>
<a NAME="E14:42_1_1_2_1_2_2"/>
<font color="Maroon" title="c13">Ab</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1_2"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="ring_1.html#E8:42_1_1_2_1_2_2"><i><font color="Green" title="E38">A17</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">Ib</font> being   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E16:42_1_1_2_1_2_2"/><i><font color="Green" title="E41">A21</font></i>: 
<font color="Maroon" title="c13">Ab</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Ib</font>
 <b>and </b><br/><a NAME="E17:42_1_1_2_1_2_2"/>
<font color="Maroon" title="c15">Ib</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a>
 ;<br/>
<a NAME="E18:42_1_1_2_1_2_2"/>
<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> </span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c4">Y</font> is <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>
 
<b>by </b><i><a class="txt" href="ring_1.html#E2:42_1_1_2"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="orders_1.html#D5" title="ORDERS_1:def.5">ORDERS_1:def 5</a></i>;<br/>
<a NAME="E19:42_1_1_2_1_2_2"/><b>then </b><i><font color="Green" title="E42">A22</font></i>: 
<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> </span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c4">Y</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p3">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> </span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c4">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_2.html#D14" title="RELAT_2:def.14">RELAT_2:def 14</a></i>;<br/>
<a NAME="E20:42_1_1_2_1_2_2"/>
 <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span>  is <a href="relat_2.html#V1" title="RELAT_2:attr.1">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" title="WELLORD2:func.1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span>  <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">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" title="E27">A2</font></i></a>, <a class="ref" href="wellord2.html#T29" title="WELLORD2:th.29">WELLORD2:29</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D9" title="RELAT_2:def.9">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" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p3">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> </span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Y</font>
 
<b>by </b><i><a class="txt" href="ring_1.html#E2:42_1_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="ring_1.html#E1:42_1_1_2"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="orders_1.html#T181" title="ORDERS_1:th.181">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">Aa</font>,<font color="Maroon" title="c13">Ab</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> </span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c4">Y</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">Ab</font>,<font color="Maroon" title="c12">Aa</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> </span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c4">Y</font> or <font color="Maroon" title="c12">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Ab</font> )
 
<b>by </b><i><a class="txt" href="ring_1.html#E19:42_1_1_2_1_2_2"><i><font color="Green" title="E42">A22</font></i></a>, <a class="txt" href="ring_1.html#E4:42_1_1_2_1_2_2"><i><font color="Green" title="E36">A15</font></i></a>, <a class="txt" href="ring_1.html#E8:42_1_1_2_1_2_2"><i><font color="Green" title="E38">A17</font></i></a>, <a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">Aa</font>,<font color="Maroon" title="c13">Ab</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span>  or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">Ab</font>,<font color="Maroon" title="c12">Aa</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span>  or <font color="Maroon" title="c12">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Ab</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E24:42_1_1_2_1_2_2"/><b>then </b>
( <font color="Maroon" title="c12">Aa</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">Ab</font> or <font color="Maroon" title="c13">Ab</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Aa</font> )
 
<b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1_2"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="ring_1.html#E8:42_1_1_2_1_2_2"><i><font color="Green" title="E38">A17</font></i></a>, <a class="txt" href="ring_1.html#E4:42_1_1_2_1_2_2"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E25:42_1_1_2_1_2_2"/>
<font color="Maroon" title="c10">a</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 <b>by </b><i><a class="txt" href="ring_1.html#E9:42_1_1_2_1_2_2"><i><font color="Green" title="E39">A18</font></i></a>, <a class="txt" href="ring_1.html#E12:42_1_1_2_1_2_2"><i><font color="Green" title="E40">A20</font></i></a>, <a class="txt" href="ring_1.html#E16:42_1_1_2_1_2_2"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="ring_1.html#E3:42_1_1_2_1_2_2"><i><font color="Green" title="E35">A14</font></i></a>, <a class="txt" href="ring_1.html#E4:42_1_1_2_1_2_2"><i><font color="Green" title="E36">A15</font></i></a>, <a class="txt" href="ring_1.html#E7:42_1_1_2_1_2_2"><i><font color="Green" title="E37">A16</font></i></a>, <a class="txt" href="ring_1.html#E8:42_1_1_2_1_2_2"><i><font color="Green" title="E38">A17</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:42_1_1_2_1_2"/><i><font color="Green" title="E35">A23</font></i>: 
<font color="Maroon" title="c9">B</font> is <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">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" title="c10">p</font>, <font color="Maroon" title="c11">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D2" title="IDEAL_1:def.2">IDEAL_1:def 2</a><br/>

<b>assume </b><a NAME="E1:42_1_1_2_1_2_3"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">Aa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:42_1_1_2_1_2_3"/><i><font color="Green" title="E36">A24</font></i>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Aa</font>
 <b>and </b><br/><a NAME="E4:42_1_1_2_1_2_3"/><i><font color="Green" title="E37">A25</font></i>: 
<font color="Maroon" title="c12">Aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:42_1_1_2_1_2_3"/>
<font color="Maroon" title="c12">Aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1_2"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="ring_1.html#E4:42_1_1_2_1_2_3"><i><font color="Green" title="E37">A25</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">Ia</font> being   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E7:42_1_1_2_1_2_3"/><i><font color="Green" title="E38">A26</font></i>: 
<font color="Maroon" title="c12">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Ia</font>
 <b>and </b><br/><a NAME="E8:42_1_1_2_1_2_3"/>
<font color="Maroon" title="c13">Ia</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a>
 ;<br/>
<a NAME="E9:42_1_1_2_1_2_3"/><i><font color="Green" title="E39">A27</font></i>: 
<font color="Maroon" title="c10">p</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Ia</font>
 
<b>by </b><i><a class="txt" href="ring_1.html#E3:42_1_1_2_1_2_3"><i><font color="Green" title="E36">A24</font></i></a>, <a class="txt" href="ring_1.html#E7:42_1_1_2_1_2_3"><i><font color="Green" title="E38">A26</font></i></a>, <a class="ref" href="ideal_1.html#D2" title="IDEAL_1:def.2">IDEAL_1:def 2</a></i>;<br/>
<a NAME="E10:42_1_1_2_1_2_3"/>
<font color="Maroon" title="c13">Ia</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">B</font>
 
<b>by </b><i><a class="txt" href="ring_1.html#E7:42_1_1_2_1_2_3"><i><font color="Green" title="E38">A26</font></i></a>, <a class="txt" href="ring_1.html#E4:42_1_1_2_1_2_3"><i><font color="Green" title="E37">A25</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E11:42_1_1_2_1_2_3"/>
<font color="Maroon" title="c10">p</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 <b>by </b><i><a class="txt" href="ring_1.html#E9:42_1_1_2_1_2_3"><i><font color="Green" title="E39">A27</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:42_1_1_2_1_2"/><i><font color="Green" title="E36">A28</font></i>: 
<font color="Maroon" title="c9">B</font> is <a href="ideal_1.html#V3" title="IDEAL_1:attr.3">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" title="c10">p</font>, <font color="Maroon" title="c11">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D3" title="IDEAL_1:def.3">IDEAL_1:def 3</a><br/>

<b>assume </b><a NAME="E1:42_1_1_2_1_2_4"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">Aa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:42_1_1_2_1_2_4"/><i><font color="Green" title="E37">A29</font></i>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Aa</font>
 <b>and </b><br/><a NAME="E4:42_1_1_2_1_2_4"/><i><font color="Green" title="E38">A30</font></i>: 
<font color="Maroon" title="c12">Aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:42_1_1_2_1_2_4"/>
<font color="Maroon" title="c12">Aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1_2"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="ring_1.html#E4:42_1_1_2_1_2_4"><i><font color="Green" title="E38">A30</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">Ia</font> being   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E7:42_1_1_2_1_2_4"/><i><font color="Green" title="E39">A31</font></i>: 
<font color="Maroon" title="c12">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Ia</font>
 <b>and </b><br/><a NAME="E8:42_1_1_2_1_2_4"/>
<font color="Maroon" title="c13">Ia</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a>
 ;<br/>
<a NAME="E9:42_1_1_2_1_2_4"/><i><font color="Green" title="E40">A32</font></i>: 
<font color="Maroon" title="c11">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Ia</font>
 
<b>by </b><i><a class="txt" href="ring_1.html#E3:42_1_1_2_1_2_4"><i><font color="Green" title="E37">A29</font></i></a>, <a class="txt" href="ring_1.html#E7:42_1_1_2_1_2_4"><i><font color="Green" title="E39">A31</font></i></a>, <a class="ref" href="ideal_1.html#D3" title="IDEAL_1:def.3">IDEAL_1:def 3</a></i>;<br/>
<a NAME="E10:42_1_1_2_1_2_4"/>
<font color="Maroon" title="c13">Ia</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">B</font>
 
<b>by </b><i><a class="txt" href="ring_1.html#E7:42_1_1_2_1_2_4"><i><font color="Green" title="E39">A31</font></i></a>, <a class="txt" href="ring_1.html#E4:42_1_1_2_1_2_4"><i><font color="Green" title="E38">A30</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E11:42_1_1_2_1_2_4"/>
<font color="Maroon" title="c11">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 <b>by </b><i><a class="txt" href="ring_1.html#E9:42_1_1_2_1_2_4"><i><font color="Green" title="E40">A32</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" title="c9">B</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a>
 ;<br/><a NAME="E2:42_1_1_2_1_2_5"/><b>then </b>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 <b>by </b><i><a class="txt" href="ring_1.html#E12:42_1_1_2_1_2"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="ring_1.html#E14:42_1_1_2_1_2"><i><font color="Green" title="E35">A23</font></i></a>, <a class="ref" href="ideal_1.html#T19" title="IDEAL_1:th.19">IDEAL_1:19</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">Aa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:42_1_1_2_1_2_5"/><i><font color="Green" title="E37">A33</font></i>: 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Aa</font>
 <b>and </b><br/><a NAME="E5:42_1_1_2_1_2_5"/><i><font color="Green" title="E38">A34</font></i>: 
<font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E6:42_1_1_2_1_2_5"/>
<font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1_2"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="ring_1.html#E5:42_1_1_2_1_2_5"><i><font color="Green" title="E38">A34</font></i></a></i>;<br/><a NAME="E7:42_1_1_2_1_2_5"/><b>then </b>
 ex <font color="Olive" title="b1">Ia</font> being  <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Ia</font> &amp; <font color="Olive" title="b1">Ia</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">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" title="E37">A33</font></i></a>, <a class="ref" href="ideal_1.html#T19" title="IDEAL_1:th.19">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" title="E37">A35</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">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" title="E36">A28</font></i></a>, <a class="txt" href="ring_1.html#E12:42_1_1_2_1_2"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="ring_1.html#E13:42_1_1_2_1_2"><i><font color="Green" title="E34">A13</font></i></a>, <a class="txt" href="ring_1.html#E14:42_1_1_2_1_2"><i><font color="Green" title="E35">A23</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c10">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E18:42_1_1_2_1_2"/><i><font color="Green" title="E38">A36</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 ;<br/><a NAME="E19:42_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c10">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c8">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1_2"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="ring_1.html#E18:42_1_1_2_1_2"><i><font color="Green" title="E38">A36</font></i></a>, <a class="txt" href="ring_1.html#E17:42_1_1_2_1_2"><i><font color="Green" title="E37">A35</font></i></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">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" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:42_1_1"/><i><font color="Green" title="E28">A37</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="orders_1.html#R6" title="ORDERS_1:pred.6">is_maximal_in</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="ring_1.html#E1:42_1_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="ring_1.html#E2:42_1_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="orders_1.html#T173" title="ORDERS_1:th.173">ORDERS_1:173</a></i>;<br/>
<a NAME="E6:42_1_1"/><i><font color="Green" title="E29">A38</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">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" title="E28">A37</font></i></a>, <a class="ref" href="orders_1.html#D11" title="ORDERS_1:def.11">ORDERS_1:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">I</font> being   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E8:42_1_1"/><i><font color="Green" title="E30">A39</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">I</font>
 <b>and </b><br/><a NAME="E9:42_1_1"/><i><font color="Green" title="E31">A40</font></i>: 
<font color="Maroon" title="c5">I</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a>
 <b>by </b><i><a class="txt" href="ring_1.html#E2:42_1_1"><i><font color="Green" title="E27">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">I</font>
;<br/>

<b>thus </b><a NAME="E10:42_1_1"/>
<font color="Maroon" title="c5">I</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a>
 <b>by </b><i><a class="txt" href="ring_1.html#E9:42_1_1"><i><font color="Green" title="E31">A40</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ring_1.html#D4" target="_self" title="RING_1:def.4">RING_1:def 4</a><br/>

<b>let </b><font color="Maroon" title="c6">J</font> be   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ring_1.html#D3" target="_self" title="RING_1:def.3">RING_1:def 3</a><br/>

<b>assume </b><a NAME="E11:42_1_1"/><i><font color="Green" title="E32">A41</font></i>: 
<font color="Maroon" title="c5">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">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" title="c6">J</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a>
 ;<br/><a NAME="E2:42_1_1_3"/><b>then </b><i><font color="Green" title="E33">A42</font></i>: 
<font color="Maroon" title="c6">J</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> 
 ;<br/><a NAME="E3:42_1_1_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c6">J</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">A</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="ring_1.html#E2:42_1_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="ring_1.html#E6:42_1_1"><i><font color="Green" title="E29">A38</font></i></a>, <a class="txt" href="ring_1.html#E8:42_1_1"><i><font color="Green" title="E30">A39</font></i></a>, <a class="txt" href="ring_1.html#E11:42_1_1"><i><font color="Green" title="E32">A41</font></i></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/><b>hence </b><a NAME="E4:42_1_1_3"/>
<font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">J</font>
 <b>by </b><i><a class="txt" href="ring_1.html#E2:42_1_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="ring_1.html#E2:42_1_1_3"><i><font color="Green" title="E33">A42</font></i></a>, <a class="txt" href="ring_1.html#E8:42_1_1"><i><font color="Green" title="E30">A39</font></i></a>, <a class="txt" href="ring_1.html#E5:42_1_1"><i><font color="Green" title="E28">A37</font></i></a>, <a class="ref" href="orders_1.html#D11" title="ORDERS_1:def.11">ORDERS_1:def 11</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:42_1_1"/>
( <font color="Maroon" title="c6">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">I</font> or  not <font color="Maroon" title="c6">J</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> )
 ;<br/>


</div>
