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

<span class="kw">set </span><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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> ;<br/>
<span class="kw">set </span><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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> ;<br/>
<a NAME="E1:39_1_1"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
 <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="subset_1.html#V1" title="SUBSET_1:attr.1">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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord2.html#T31" title="WELLORD2:th.31">WELLORD2:31</a></span>;<br/>
<a NAME="E2:39_1_1"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
 <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="subset_1.html#V1" title="SUBSET_1:attr.1">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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>;<br/>
<a NAME="E3:39_1_1"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<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="subset_1.html#V1" title="SUBSET_1:attr.1">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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="orders_1.html#D9" title="ORDERS_1:def.9">ORDERS_1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span>  or  not <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="subset_1.html#V1" title="SUBSET_1:attr.1">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="orders_1.html#V3" title="ORDERS_1:attr.3">being_linear-order</a>  or  ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</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>2</sub></font>,<font color="Olive">b<sub>1</sub></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="b3">A</font> where <font color="Olive" title="b3">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="b3">A</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span>  ) ) ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:39_1_1_1"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
<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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> 
 <span class="kw">and </span><br/><a NAME="E2:39_1_1_1"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
<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="subset_1.html#V1" title="SUBSET_1:attr.1">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="orders_1.html#V3" title="ORDERS_1:attr.3">being_linear-order</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</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>2</sub></font>,<font color="Olive">b<sub>1</sub></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="b3">A</font> where <font color="Olive" title="b3">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="b3">A</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span>  ) ) )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:39_1_1_1_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="39_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:39_1_1_1_1_1"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
<font color="Maroon" title="c4">Y</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</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>2</sub></font>,<font color="Olive">b<sub>1</sub></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="b3">A</font> where <font color="Olive" title="b3">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="b3">A</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span>  ) ) )</span><br/><div class="add"><span class="kw">take </span><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="struct_0.html#K4" title="STRUCT_0:func.4">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span>  &amp; ( 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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span>  ) ) )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_1_1_1_1_1_1"/>
not <font color="Maroon" title="c5">x</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:39_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#D7" title="SUBSET_1:def.7">SUBSET_1:def 7</a></span>;<br/><a NAME="E3:39_1_1_1_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="struct_0.html#K4" title="STRUCT_0:func.4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ideal_1.html#T47" title="IDEAL_1:th.47">IDEAL_1:47</a></span>;<br/><a NAME="E4:39_1_1_1_1_1_1"/><span class="kw">then </span>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:39_1_1_1_1_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:39_1_1_1_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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span>  )</span><br/><span class="kw">thus </span><a NAME="E4:39_1_1_1_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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_1_1_1_1"><span class="lab"><font color="Green" title="E29">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:39_1_1_1_1_2"/>
not <font color="Maroon" title="c4">Y</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</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>2</sub></font>,<font color="Olive">b<sub>1</sub></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="b3">A</font> where <font color="Olive" title="b3">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="b3">A</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span>  ) ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:39_1_1_1_1_2"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c6">x</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span>  &amp; ( 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="c6">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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span>  ) ) )</span><br/><a NAME="E4:39_1_1_1_1_2"/>
<font color="Maroon" title="c6">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="39_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">x</font> or <font color="Maroon" title="c7">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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_1_1_1_1_2_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">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></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:39_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font>
 <span class="kw">and </span><br/><a NAME="E4:39_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
<font color="Maroon" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E5:39_1_1_1_1_2_1"/>
<font color="Maroon" title="c8">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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_1_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E4:39_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E31">A10</font></span></a></span>;<br/>
<a NAME="E6:39_1_1_1_1_2_1"/><span class="kw">then </span>
 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="c8">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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  )
 
;<br/>
<span class="kw">hence </span><a NAME="E7:39_1_1_1_1_2_1"/>
<font color="Maroon" title="c7">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E30">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c6">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="E6:39_1_1_1_1_2"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
<font color="Maroon" title="c7">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="39_1_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c9">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ideal_1.html#D3" title="IDEAL_1:def.3">IDEAL_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> or <font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_1_1_1_1_2_2"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">Aa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:39_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E31">A12</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Aa</font>
 <span class="kw">and </span><br/><a NAME="E4:39_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E32">A13</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E5:39_1_1_1_1_2_2"/>
<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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_1_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E4:39_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E32">A13</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">Ia</font> being   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E7:39_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E33">A14</font></span>: 
<font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Ia</font>
 <span class="kw">and </span><br/><a NAME="E8:39_1_1_1_1_2_2"/>
<font color="Maroon" title="c11">Ia</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> 
 ;<br/>
<a NAME="E9:39_1_1_1_1_2_2"/>
( <font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ia</font> &amp; <font color="Maroon" title="c11">Ia</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E4:39_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E7:39_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="ref" href="ideal_1.html#D3" title="IDEAL_1:def.3">IDEAL_1:def 3</a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:39_1_1_1_1_2_2"/>
<font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:39_1_1_1_1_2"/><span class="lab"><font color="Green" title="E31">A15</font></span>: 
<font color="Maroon" title="c7">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="39_1_1_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c9">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ideal_1.html#D2" title="IDEAL_1:def.2">IDEAL_1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> or <font color="Maroon" title="c8">p</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_1_1_1_1_2_3"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">p</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">Aa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:39_1_1_1_1_2_3"/><span class="lab"><font color="Green" title="E32">A16</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Aa</font>
 <span class="kw">and </span><br/><a NAME="E4:39_1_1_1_1_2_3"/><span class="lab"><font color="Green" title="E33">A17</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E5:39_1_1_1_1_2_3"/>
<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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_1_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E4:39_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E33">A17</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">Ia</font> being   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E7:39_1_1_1_1_2_3"/><span class="lab"><font color="Green" title="E34">A18</font></span>: 
<font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Ia</font>
 <span class="kw">and </span><br/><a NAME="E8:39_1_1_1_1_2_3"/>
<font color="Maroon" title="c11">Ia</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> 
 ;<br/>
<a NAME="E9:39_1_1_1_1_2_3"/>
( <font color="Maroon" title="c8">p</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ia</font> &amp; <font color="Maroon" title="c11">Ia</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E4:39_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="txt" href="#E7:39_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="ref" href="ideal_1.html#D2" title="IDEAL_1:def.2">IDEAL_1:def 2</a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:39_1_1_1_1_2_3"/>
<font color="Maroon" title="c8">p</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E32">A19</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_2_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_1_1_1_1_2_4"/>
not <font color="Maroon" title="c7">B</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:39_1_1_1_1_2_4"/><span class="kw">then </span>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="ref" href="ideal_1.html#T19" title="IDEAL_1:th.19">IDEAL_1:19</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">Aa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:39_1_1_1_1_2_4"/><span class="lab"><font color="Green" title="E33">A20</font></span>: 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Aa</font>
 <span class="kw">and </span><br/><a NAME="E5:39_1_1_1_1_2_4"/><span class="lab"><font color="Green" title="E34">A21</font></span>: 
<font color="Maroon" title="c8">Aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E6:39_1_1_1_1_2_4"/>
<font color="Maroon" title="c8">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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_1_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E5:39_1_1_1_1_2_4"><span class="lab"><font color="Green" title="E34">A21</font></span></a></span>;<br/><a NAME="E7:39_1_1_1_1_2_4"/><span class="kw">then </span>
 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="c8">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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  )
 ;<br/><span class="kw">hence </span><a NAME="E8:39_1_1_1_1_2_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_1_1_1_2_4"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="ref" href="ideal_1.html#T19" title="IDEAL_1:th.19">IDEAL_1:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E9:39_1_1_1_1_2"/><span class="lab"><font color="Green" title="E33">A22</font></span>: 
<font color="Maroon" title="c7">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="39_1_1_1_1_2_5"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:39_1_1_1_1_2_5"/>
 <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="subset_1.html#V1" title="SUBSET_1:attr.1">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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="wellord2.html#T29" title="WELLORD2:th.29">WELLORD2:29</a></span>;<br/>
<a NAME="E2:39_1_1_1_1_2_5"/><span class="kw">then </span>
 <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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span>  is  <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D9" title="RELAT_2:def.9">RELAT_2:def 9</a></span>;<br/>
<a NAME="E3:39_1_1_1_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A23</font></span>: 
 <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="subset_1.html#V1" title="SUBSET_1:attr.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E1:39_1_1_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="orders_1.html#T181" title="ORDERS_1:th.181">ORDERS_1:181</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ideal_1.html#D1" title="IDEAL_1:def.1">IDEAL_1:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> or  not <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> or <font color="Maroon" title="c8">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> )</span><br/>

<div><span class="lab"><font color="Green" title="E35">A24</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_1_2_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">A</font> be   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> &amp; <font color="Maroon" title="c10">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> implies <font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> )</span><br/><span class="kw">assume </span><a NAME="E1:39_1_1_1_1_2_5_1"/>
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> implies <font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> )</span><br/><a NAME="E2:39_1_1_1_1_2_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A25</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ideal_1.html#D1" title="IDEAL_1:def.1">IDEAL_1:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E3:39_1_1_1_1_2_5_1"/>
<font color="Maroon" title="c10">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font></span><br/><span class="kw">hence </span><a NAME="E4:39_1_1_1_1_2_5_1"/>
<font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1_1_1_2_5_1"><span class="lab"><font color="Green" title="E36">A25</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E5:39_1_1_1_1_2_5"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> or <font color="Maroon" title="c8">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">Aa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:39_1_1_1_1_2_5"/><span class="lab"><font color="Green" title="E36">A26</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Aa</font>
 <span class="kw">and </span><br/><a NAME="E8:39_1_1_1_1_2_5"/><span class="lab"><font color="Green" title="E37">A27</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E9:39_1_1_1_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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_1_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E8:39_1_1_1_1_2_5"><span class="lab"><font color="Green" title="E37">A27</font></span></a></span>;<br/>
<a NAME="E10:39_1_1_1_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A28</font></span>: 
 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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  )
 
;<br/>
<span class="kw">assume </span><a NAME="E11:39_1_1_1_1_2_5"/>
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">Ab</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E13:39_1_1_1_1_2_5"/><span class="lab"><font color="Green" title="E39">A29</font></span>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ab</font>
 <span class="kw">and </span><br/><a NAME="E14:39_1_1_1_1_2_5"/><span class="lab"><font color="Green" title="E40">A30</font></span>: 
<font color="Maroon" title="c11">Ab</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E15:39_1_1_1_1_2_5"/>
<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="subset_1.html#V1" title="SUBSET_1:attr.1">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="orders_1.html#D5" title="ORDERS_1:def.5">ORDERS_1:def 5</a></span>;<br/>
<a NAME="E16:39_1_1_1_1_2_5"/><span class="kw">then </span>
<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="subset_1.html#V1" title="SUBSET_1:attr.1">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="subset_1.html#V1" title="SUBSET_1:attr.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D14" title="RELAT_2:def.14">RELAT_2:def 14</a></span>;<br/>
<a NAME="E17:39_1_1_1_1_2_5"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">Aa</font>,<font color="Maroon" title="c11">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="subset_1.html#V1" title="SUBSET_1:attr.1">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="c11">Ab</font>,<font color="Maroon" title="c10">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="subset_1.html#V1" title="SUBSET_1:attr.1">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="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Ab</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39_1_1_1_1_2_5"><span class="lab"><font color="Green" title="E37">A27</font></span></a>, <a class="txt" href="#E14:39_1_1_1_1_2_5"><span class="lab"><font color="Green" title="E40">A30</font></span></a>, <a class="txt" href="#E3:39_1_1_1_1_2_5"><span class="lab"><font color="Green" title="E34">A23</font></span></a>, <a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a></span>;<br/>
<a NAME="E18:39_1_1_1_1_2_5"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">Aa</font>,<font color="Maroon" title="c11">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="subset_1.html#V1" title="SUBSET_1:attr.1">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="c11">Ab</font>,<font color="Maroon" title="c10">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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span>  or <font color="Maroon" title="c10">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Ab</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E19:39_1_1_1_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A31</font></span>: 
( <font color="Maroon" title="c10">Aa</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Ab</font> or <font color="Maroon" title="c11">Ab</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Aa</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_1_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E8:39_1_1_1_1_2_5"><span class="lab"><font color="Green" title="E37">A27</font></span></a>, <a class="txt" href="#E14:39_1_1_1_1_2_5"><span class="lab"><font color="Green" title="E40">A30</font></span></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>;<br/>
<a NAME="E20:39_1_1_1_1_2_5"/>
<font color="Maroon" title="c11">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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_1_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E14:39_1_1_1_1_2_5"><span class="lab"><font color="Green" title="E40">A30</font></span></a></span>;<br/>
<a NAME="E21:39_1_1_1_1_2_5"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">Ib</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="c11">Ab</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Ib</font> &amp; <font color="Olive" title="b1">Ib</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  )
 
;<br/>
<span class="kw">hence </span><a NAME="E22:39_1_1_1_1_2_5"/>
<font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_1_1_1_1_2_5"><span class="lab"><font color="Green" title="E36">A26</font></span></a>, <a class="txt" href="#E8:39_1_1_1_1_2_5"><span class="lab"><font color="Green" title="E37">A27</font></span></a>, <a class="txt" href="#E13:39_1_1_1_1_2_5"><span class="lab"><font color="Green" title="E39">A29</font></span></a>, <a class="txt" href="#E14:39_1_1_1_1_2_5"><span class="lab"><font color="Green" title="E40">A30</font></span></a>, <a class="txt" href="#E4:39_1_1_1_1_2_5"><span class="lab"><font color="Green" title="E35">A24</font></span></a>, <a class="txt" href="#E10:39_1_1_1_1_2_5"><span class="lab"><font color="Green" title="E38">A28</font></span></a>, <a class="txt" href="#E19:39_1_1_1_1_2_5"><span class="lab"><font color="Green" title="E41">A31</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E10:39_1_1_1_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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_1_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E3:39_1_1_1_1_2"><span class="lab"><font color="Green" title="E29">A8</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">A</font> being   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E12:39_1_1_1_1_2"/><span class="lab"><font color="Green" title="E34">A32</font></span>: 
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">and </span><br/><a NAME="E13:39_1_1_1_1_2"/>
<font color="Maroon" title="c8">A</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> 
 ;<br/><a NAME="E14:39_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">q</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><a NAME="E15:39_1_1_1_1_2"/><span class="kw">then </span>
not <font color="Maroon" title="c7">B</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1_1_1_1_2"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E12:39_1_1_1_1_2"><span class="lab"><font color="Green" title="E34">A32</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E16:39_1_1_1_1_2"/><span class="lab"><font color="Green" title="E35">A33</font></span>: 
<font color="Maroon" title="c6">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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_1_1_1_2"><span class="lab"><font color="Green" title="E33">A22</font></span></a>, <a class="txt" href="#E7:39_1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="txt" href="#E6:39_1_1_1_1_2"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="txt" href="#E8:39_1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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="c6">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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">y</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="Maroon" title="c9">y</font>,<font color="Maroon" title="c6">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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span>  )</span><br/><span class="kw">assume </span><a NAME="E17:39_1_1_1_1_2"/><span class="lab"><font color="Green" title="E36">A34</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c6">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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> </span><br/><a NAME="E18:39_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/><span class="kw">hence </span><a NAME="E19:39_1_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c6">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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_1_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E16:39_1_1_1_1_2"><span class="lab"><font color="Green" title="E35">A33</font></span></a>, <a class="txt" href="#E17:39_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A34</font></span></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E4:39_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="subset_1.html#V1" title="SUBSET_1:attr.1">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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span>  &amp;  <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="subset_1.html#V1" title="SUBSET_1:attr.1">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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord2.html#T29" title="WELLORD2:th.29">WELLORD2:29</a>, <a class="ref" href="wellord2.html#T30" title="WELLORD2:th.30">WELLORD2:30</a></span>;<br/>
<a NAME="E5:39_1_1"/><span class="kw">then </span>
 <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="subset_1.html#V1" title="SUBSET_1:attr.1">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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="orders_1.html#D7" title="ORDERS_1:def.7">ORDERS_1:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:39_1_1"/><span class="lab"><font color="Green" title="E27">A35</font></span>: 
<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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E3:39_1_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="orders_1.html#T173" title="ORDERS_1:th.173">ORDERS_1:173</a></span>;<br/>
<a NAME="E8:39_1_1"/><span class="lab"><font color="Green" title="E28">A36</font></span>: 
<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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_1_1"><span class="lab"><font color="Green" title="E27">A35</font></span></a>, <a class="ref" href="orders_1.html#D11" title="ORDERS_1:def.11">ORDERS_1:def 11</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E10:39_1_1"/><span class="lab"><font color="Green" title="E29">A37</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">I</font>
 <span class="kw">and </span><br/><a NAME="E11:39_1_1"/><span class="lab"><font color="Green" title="E30">A38</font></span>: 
<font color="Maroon" title="c5">I</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">I</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">I</font> is  <a href="ring_1.html#V4" title="RING_1:attr.4">maximal</a> </span><br/>

<span class="kw">thus </span><a NAME="E12:39_1_1"/>
<font color="Maroon" title="c5">I</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:39_1_1"><span class="lab"><font color="Green" title="E30">A38</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ring_1.html#D4" target="_self" title="RING_1:def.4">RING_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">I</font> is  <a href="ring_1.html#V3" title="RING_1:attr.3">quasi-maximal</a> </span><br/>

<span class="kw">let </span><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>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ring_1.html#D3" target="_self" title="RING_1:def.3">RING_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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> or <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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E13:39_1_1"/><span class="lab"><font color="Green" title="E31">A39</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_1_1_2"/>
<font color="Maroon" title="c6">J</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">J</font></span><br/><a NAME="E2:39_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A40</font></span>: 
<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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> 
 ;<br/><a NAME="E3:39_1_1_2"/><span class="kw">then </span>
<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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E8:39_1_1"><span class="lab"><font color="Green" title="E28">A36</font></span></a>, <a class="txt" href="#E10:39_1_1"><span class="lab"><font color="Green" title="E29">A37</font></span></a>, <a class="txt" href="#E13:39_1_1"><span class="lab"><font color="Green" title="E31">A39</font></span></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:39_1_1_2"/>
<font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">J</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E7:39_1_1"><span class="lab"><font color="Green" title="E27">A35</font></span></a>, <a class="txt" href="#E10:39_1_1"><span class="lab"><font color="Green" title="E29">A37</font></span></a>, <a class="txt" href="#E2:39_1_1_2"><span class="lab"><font color="Green" title="E32">A40</font></span></a>, <a class="ref" href="orders_1.html#D11" title="ORDERS_1:def.11">ORDERS_1:def 11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E15:39_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="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
