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<b>let </b><font color="Maroon">R</font> be   <a href="vectsp_2.html#NM1">comRing</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>



<b>set </b><font color="Maroon">M</font> = <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">r</font></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : verum </span>}</span> ;<br/>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_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="E1:41_1"/>
<font color="Maroon">u</font> <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">r</font></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:41_1"/><i><font color="Green">E42</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> <font color="Maroon">r</font>
 ;<br/><b>thus </b><a NAME="E4:41_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>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E2:41"/>
<font color="Maroon">a</font> <a href="group_1.html#K10">*</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> <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">r</font></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : verum </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">M</font> = <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">r</font></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : verum </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> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">M</font> = <font color="Maroon">M</font> 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/>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/><b>assume </b><a NAME="E1:41_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 ;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:41_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><b>consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E5:41_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/><a NAME="E6:41_2"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</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">r</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a></i>;<br/><b>hence </b><a NAME="E7:41_2"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E77</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/><b>assume </b><a NAME="E1:41_3"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:41_3"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E4:41_3"/>
<font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s</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">r</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/><b>hence </b><a NAME="E5:41_3"/>
<font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/><b>assume </b><a NAME="E1:41_4"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:41_4"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E4:41_4"/>
<font color="Maroon">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/><b>hence </b><a NAME="E5:41_4"/>
<font color="Maroon">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:41"/>
<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">r</font></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : verum </span>}</span>  is   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font>
 <b>by </b><i>, , , , </i>;<br/>


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