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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#L1">multLoopStr</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#V3" target="_self">right-ideal</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">J</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_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="E1:141_1"/><i><font color="Green">E41</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:141_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u'</font> <a href="ideal_1.html#K10" target="_self">*</a> <font color="Maroon">J</font>
 ;<br/><a NAME="E2:141_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">u'</font> <a href="group_1.html#K1">*</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E4:141_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> )
 ;<br/><b>thus </b><a NAME="E5:141_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#D5" target="_self">Def5</a>, , </i>;<br/></div><b>end;</b></div><a NAME="E4:141_1"/><b>then </b>
<font color="Maroon">u'</font> <a href="ideal_1.html#K10" target="_self">*</a> <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="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E5:141_1"/><b>then </b>
<font color="Maroon">u'</font> <a href="hidden.html#R2">in</a> <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> : <font color="Olive">a</font> <a href="ideal_1.html#K10" target="_self">*</a> <font color="Maroon">J</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E6:141_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="ideal_1.html#K16" target="_self">%</a> <font color="Maroon">J</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:141"/>
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> <a href="ideal_1.html#K16" target="_self">%</a> <font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


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