<?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> :  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">a</font> <a href="binom.html#K2">|^</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> </span>}</span> ;<br/>
<a NAME="E1:150_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> :  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">a</font> <a href="binom.html#K2">|^</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="ideal_1.html#K17" target="_self">sqrt</a> <font color="Maroon">I</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> :  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">a</font> <a href="binom.html#K2">|^</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</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="E3:150_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="150_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="E1:150_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x'</font>, <font color="Maroon">y</font> = <font color="Maroon">y'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ;<br/>
<b>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="E4:150_1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a</font> <a href="binom.html#K2">|^</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:150_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> <a href="binom.html#K2">|^</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E7:150_1_1_1"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="binom.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="ideal_1.html#T14" target="_self">Th14</a>, </i>;<br/>
<a NAME="E8:150_1_1_1"/>
<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="binom.html#K2">|^</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="binom.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binom.html#K2">|^</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="binom.html#T10">BINOM:10</a></i>;<br/>
<b>hence </b><a NAME="E9:150_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#T13" target="_self">Th13</a>, <a class="ref" href="ideal_1.html#T17" target="_self">Th17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:150_1_1"/>
 <a href="ideal_1.html#K17" target="_self">sqrt</a> <font color="Maroon">I</font> is <a href="ideal_1.html#V2" target="_self">left-ideal</a>
 <b>by </b><i/>;<br/>


</div>
