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<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a> <a href="vectsp_1.html#V8">left_unital</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">I</font> be   <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="105_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="E1:105_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</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="E3:105_1"/>
<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> <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u'</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>;<br/><a NAME="E4:105_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"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="group_1.html#K1">*</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">I</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T10" target="_self">Th10</a></i>;<br/><b>hence </b><a NAME="E5:105_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="ideal_1.html#K10" target="_self">*</a> <font color="Maroon">I</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2"><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="E1:105_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="ideal_1.html#K10" target="_self">*</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:105_2"/><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"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="group_1.html#K1">*</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">I</font> </span>}</span> 
 ;<br/><a NAME="E3:105_2"/><b>then </b>
 ex <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <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> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 ;<br/><b>hence </b><a NAME="E4:105_2"/>
<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="vectsp_1.html#D19">VECTSP_1:def 19</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:105"/>
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="ideal_1.html#K10" target="_self">*</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T6" target="_self">Th6</a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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