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

<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a> <a href="vectsp_1.html#V6">right_unital</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ideal_1.html#V2" target="_self">left-ideal</a> <a href="ideal_1.html#V3" target="_self">right-ideal</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>;<br/>



<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">I</font> is <a href="ideal_1.html#V5" target="_self">proper</a>
 ;<br/><a NAME="E2:44_1"/><i><font color="Green">E43</font></i>: 
not  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_1_1"/><i><font color="Green">E44</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:44_1_1"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_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="E1:44_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/><b>then reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ;<br/><a NAME="E3:44_1_1_1"/>
<font color="Maroon">u'</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u'</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>;<br/><b>hence </b><a NAME="E4:44_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E4:44_1_1"/><b>then </b>
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E5:44_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:44_1"/>
for <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  st <font color="Olive">u</font> is <a href="gcd_1.html#V1">unital</a> holds <br/> not <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:44_1_2"/>
<font color="Maroon">u</font> is <a href="gcd_1.html#V1">unital</a>
 ;<br/>

<a NAME="E2:44_1_2"/><b>then </b>
<font color="Maroon">u</font> <a href="gcd_1.html#R1">divides</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D2">GCD_1:def 2</a></i>;<br/>
<a NAME="E3:44_1_2"/><b>then </b>
 ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D1">GCD_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:44_1_2"/>
not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_2"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  st <font color="Olive">u</font> is <a href="gcd_1.html#V1">unital</a> holds <br/> not <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:44_2"/>
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>;<br/><a NAME="E3:44_2"/><b>then </b>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="gcd_1.html#R1">divides</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D1">GCD_1:def 1</a></i>;<br/><a NAME="E4:44_2"/><b>then </b>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> is <a href="gcd_1.html#V1">unital</a>
 <b>by </b><i><a class="ref" href="gcd_1.html#D2">GCD_1:def 2</a></i>;<br/><a NAME="E5:44_2"/><b>then </b>
<font color="Maroon">I</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:44_2"/>
<font color="Maroon">I</font> is <a href="ideal_1.html#V5" target="_self">proper</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:44"/>
( <font color="Maroon">I</font> is <a href="ideal_1.html#V5" target="_self">proper</a> iff for <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  st <font color="Olive">u</font> is <a href="gcd_1.html#V1">unital</a> holds <br/> not <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 <b>by </b><i/>;<br/>


</div>
