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

<b>set </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : <font color="Olive">a</font> <a href="ideal_1.html#K10" target="_self">*</a> <font color="Maroon">J</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> </span>}</span> ;<br/>
<a NAME="E1:140_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : <font color="Olive">a</font> <a href="ideal_1.html#K10" target="_self">*</a> <font color="Maroon">J</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="ideal_1.html#K16" target="_self">%</a> <font color="Maroon">J</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : <font color="Olive">a</font> <a href="ideal_1.html#K10" target="_self">*</a> <font color="Maroon">J</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> ;<br/>
<a NAME="E3:140_1_1"/>
for <font color="Olive">y</font>, <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> holds <br/><font color="Olive">y</font> <a href="group_1.html#K10">*</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="140_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

<b>assume </b><a NAME="E1:140_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:140_1_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="ideal_1.html#K10" target="_self">*</a> <font color="Maroon">J</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="140_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:140_1_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="ideal_1.html#K10" target="_self">*</a> <font color="Maroon">J</font>
 ;<br/><a NAME="E2:140_1_1_1_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E4:140_1_1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> )
 ;<br/><a NAME="E5:140_1_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/><a NAME="E6:140_1_1_1_1"/>
<font color="Maroon">y</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:140_1_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> </span>}</span> 
 ;<br/><a NAME="E8:140_1_1_1_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="ideal_1.html#K10" target="_self">*</a> <font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T13" target="_self">Th13</a></i>;<br/><b>hence </b><a NAME="E9:140_1_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:140_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="ideal_1.html#K10" target="_self">*</a> <font color="Maroon">J</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:140_1_1_1"/>
<font color="Maroon">y</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#D5" target="_self">Def5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:140_1_1"/>
<font color="Maroon">I</font> <a href="ideal_1.html#K16" target="_self">%</a> <font color="Maroon">J</font> is <a href="ideal_1.html#V2" target="_self">left-ideal</a>
 <b>by </b><i/>;<br/>


</div>
