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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">I</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>, <font color="Maroon">J</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="108_1_1_1_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="E1:108_1_1_1_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>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <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="E3:108_1_1_1_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="rlvect_1.html#K4">+</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> )
 ;<br/><b>thus </b><a NAME="E4:108_1_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T10" target="_self">Th10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_1_1_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="E1:108_1_1_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">I</font>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <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="E3:108_1_1_1_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 ;<br/><b>thus </b><a NAME="E4:108_1_1_1_2"/>
<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><a class="ref" href="ideal_1.html#T10" target="_self">Th10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:108_1_1_1"/>
<font color="Maroon">I</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Maroon">J</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="ideal_1.html#K11" target="_self">+</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/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:108_1_1"/>
for <font color="Olive">I</font>, <font color="Olive">J</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> holds  <font color="Olive">I</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Olive">J</font> <a href="hidden.html#R1">=</a> <font color="Olive">J</font> <a href="ideal_1.html#K11" target="_self">+</a> <font color="Olive">I</font>
 ;<br/>


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