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

<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">D</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> holds   <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">D</font> is   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">R</font></span><br/><b>let </b><font color="Maroon" title="c2">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> is   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">R</font></span><br/>



<a NAME="E1:2"/>
<font color="Maroon" title="c2">D</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 <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E2:2"/><b>then </b><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c2">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E3:2"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c2">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</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="2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">d</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> or <font color="Maroon" title="c3">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</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/>

<b>assume </b><a NAME="E1:2_1"/>
<font color="Maroon" title="c3">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</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/>

<a NAME="E2:2_1"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="#E2:2"><i><font color="Green" title="E3">A1</font></i></a></i>;<br/>
<a NAME="E3:2_1"/><b>then </b>
 ex <font color="Olive" title="b1">I</font> being  <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I</font>
 
;<br/>
<b>hence </b><a NAME="E4:2_1"/>
<font color="Maroon" title="c3">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">d</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">D</font>;<br/>
<a NAME="E5:2"/>
<font color="Maroon" title="c3">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 
;<br/>
<a NAME="E6:2"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="#E2:2"><i><font color="Green" title="E3">A1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">I</font> being   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E8:2"/><i><font color="Green" title="E5">A3</font></i>: 
<font color="Maroon" title="c3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">I</font>
 <b>and </b><br/><a NAME="E9:2"/>
verum
 ;<br/>
<b>consider </b><font color="Maroon" title="c5">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">I</font>;<br/>
<a NAME="E11:2"/><i><font color="Green" title="E6">A4</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="txt" href="#E8:2"><i><font color="Green" title="E5">A3</font></i></a></i>;<br/>
<a NAME="E12:2"/><i><font color="Green" title="E7">A5</font></i>: 
for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> holds <br/><font color="Olive" title="b1">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> implies <font color="Maroon" title="c6">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )</span><br/>

<b>assume </b><a NAME="E1:2_2"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a></span><br/>

<a NAME="E2:2_2"/><b>then </b>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="#E2:2"><i><font color="Green" title="E3">A1</font></i></a></i>;<br/>
<a NAME="E3:2_2"/><b>then </b>
 ex <font color="Olive" title="b1">I</font> being  <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I</font>
 
;<br/>
<b>hence </b><a NAME="E4:2_2"/>
<font color="Maroon" title="c6">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E13:2"/><i><font color="Green" title="E8">A6</font></i>: 
for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Olive" title="b2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> holds <br/> ex <font color="Olive" title="b3">Z</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b3">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Olive" title="b1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">Z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">X</font>, <font color="Maroon" title="c7">Y</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> implies  ex <font color="Olive" title="b1">Z</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c6">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z</font> ) )</span><br/>

<b>assume </b><a NAME="E1:2_3"/><i><font color="Green" title="E9">A7</font></i>: 
( <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">Z</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c6">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z</font> )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c8">X1</font> = <font color="Maroon" title="c6">X</font>, <font color="Maroon" title="c9">Y1</font> = <font color="Maroon" title="c7">Y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon" title="c10">Z1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:2_3"/><i><font color="Green" title="E10">A8</font></i>: 
( <font color="Maroon" title="c10">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c8">X1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">Z1</font> &amp; <font color="Maroon" title="c9">Y1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">Z1</font> )
 <b>by </b><i><a class="txt" href="#E1:2_3"><i><font color="Green" title="E9">A7</font></i></a>, <a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">WAYBEL_0:def 1</a></i>;<br/>
<a NAME="E5:2_3"/>
<font color="Maroon" title="c10">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="#E2:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="#E4:2_3"><i><font color="Green" title="E10">A8</font></i></a></i>;<br/>
<a NAME="E6:2_3"/><b>then </b>
 ex <font color="Olive" title="b1">I</font> being  <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c10">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">Z</font> = <font color="Maroon" title="c10">Z1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c11">Z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c6">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Z</font> )</span><br/>

<b>thus </b><a NAME="E8:2_3"/>
<font color="Maroon" title="c11">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 <b>by </b><i><a class="txt" href="#E4:2_3"><i><font color="Green" title="E10">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Z</font></span><br/>

<a NAME="E9:2_3"/>
( <font color="Maroon" title="c8">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Z1</font> &amp; <font color="Maroon" title="c9">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Z1</font> )
 
<b>by </b><i><a class="txt" href="#E4:2_3"><i><font color="Green" title="E10">A8</font></i></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/>
<b>hence </b><a NAME="E10:2_3"/>
<font color="Maroon" title="c6">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Z</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E14:2"/>
for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> holds <br/><font color="Olive" title="b1">X</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> implies <font color="Maroon" title="c6">X</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> )</span><br/>

<b>assume </b><a NAME="E1:2_4"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">X</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a></span><br/>

<a NAME="E2:2_4"/><b>then </b>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="#E2:2"><i><font color="Green" title="E3">A1</font></i></a></i>;<br/>
<a NAME="E3:2_4"/><b>then </b>
 ex <font color="Olive" title="b1">I</font> being  <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I</font>
 
;<br/>
<b>hence </b><a NAME="E4:2_4"/>
<font color="Maroon" title="c6">X</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:2"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> is   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="#E3:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="#E11:2"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="#E12:2"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="#E13:2"><i><font color="Green" title="E8">A6</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a>, <a class="ref" href="waybel_0.html#T26" title="WAYBEL_0:th.26">WAYBEL_0:26</a>, <a class="ref" href="waybel_0.html#T46" title="WAYBEL_0:th.46">WAYBEL_0:46</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
