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

<b>let </b><font color="Maroon">R</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">I</font> be   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font>;<br/>



<b>set </b><font color="Maroon">E</font> =  <a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font>;<br/>
<a NAME="E1:39"/><i><font color="Green">E42</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="ring_1.html#NK3" target="_self">/</a> <font color="Maroon">I</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:39"/><i><font color="Green">E47</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<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> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="ring_1.html#NK3" target="_self">/</a> <font color="Maroon">I</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E3:39"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">R</font> <a href="ring_1.html#NK3" target="_self">/</a> <font color="Maroon">I</font> is <a href="vectsp_1.html#V9">Field-like</a>
 ;<br/>

<b>given </b><font color="Maroon">J</font> being   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font><b> such that </b><a NAME="E4:39"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">J</font>
 <b>and </b><br/><a NAME="E5:39"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">J</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">I</font>
 <b>and </b><br/><a NAME="E6:39"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">J</font> is <a href="ideal_1.html#V5">proper</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="ring_1.html#D3" target="_self">RING_1:def 3</a><br/>

<a NAME="E7:39"/>
not <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="ring_1.html#T2" target="_self">Th2</a>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:39"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 <b>and </b><br/><a NAME="E10:39"/><i><font color="Green">E65</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x</font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="ring_1.html#NK3" target="_self">/</a> <font color="Maroon">I</font></span>)</span> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="ring_1.html#NK3" target="_self">/</a> <font color="Maroon">I</font></span>)</span>
 ;<br/><a NAME="E2:39_1"/><b>then </b>
<font color="Maroon">a</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:39_1"/>
contradiction
 <b>by </b><i><a class="ref" href="ring_1.html#D1" target="_self">Def1</a>, <a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="ring_1.html#NK3" target="_self">/</a> <font color="Maroon">I</font></span>)</span><b> such that </b><br/><a NAME="E15:39"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="ring_1.html#NK3" target="_self">/</a> <font color="Maroon">I</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="vectsp_1.html#D20">VECTSP_1:def 20</a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E17:39"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">b</font>
 <b>by </b><i/>;<br/>
<a NAME="E18:39"/>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E19:39"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K6">-</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#R2">in</a> <font color="Maroon">I</font>
 
<b>by </b><i>, <a class="ref" href="ring_1.html#D2" target="_self">Def2</a>, </i>;<br/>
<a NAME="E20:39"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 
<b>by </b><i>, <a class="ref" href="ideal_1.html#D3">IDEAL_1:def 3</a></i>;<br/>
<a NAME="E21:39"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E22:39"/><b>then </b>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 
<b>by </b><i>, <a class="ref" href="ring_1.html#T2" target="_self">Th2</a>, <a class="ref" href="ring_1.html#D3" target="_self">Def3</a>, <a class="ref" href="ideal_1.html#T15">IDEAL_1:15</a></i>;<br/>
<b>hence </b><a NAME="E23:39"/>
contradiction
 <b>by </b><i><a class="ref" href="ring_1.html#T3" target="_self">Th3</a>, <a class="ref" href="ideal_1.html#T19">IDEAL_1:19</a></i>;<br/>


</div>
