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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#V4">right-distributive</a> <a href="algstr_1.html#V1">left_zeroed</a> <a href="algstr_1.html#V3">add-right-cancelable</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#V3" target="_self">right-ideal</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>, <font color="Maroon">J</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/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_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="E1:114_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/><b>then reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">J</font> ;<br/><a NAME="E3:114_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font>
 <b>by </b><i/>;<br/><a NAME="E4:114_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <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="Olive">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> ) </span>}</span> 
 ;<br/><a NAME="E5:114_1"/>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span> <a href="rlvect_1.html#K2">+</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="algstr_1.html#D5">ALGSTR_1:def 5</a></i>;<br/><b>hence </b><a NAME="E6:114_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">J</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:114"/>
<font color="Maroon">J</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> <a href="ideal_1.html#K11" 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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