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<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="algstr_1.html#V1">left_zeroed</a> <a href="binom.html#V1">add-cancelable</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">F</font> be  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><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:92_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 ;<br/><b>then consider </b><font color="Maroon">lc</font> being    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E3:92_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font>
 <b>by </b><i/>;<br/><a NAME="E4:92_1_1"/>
<font color="Maroon">lc</font> is    <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E5:92_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="ideal_1.html#K8" target="_self">-LeftIdeal</a> 
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:92_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="ideal_1.html#K8" target="_self">-LeftIdeal</a> 
 ;<br/><b>then consider </b><font color="Maroon">lc</font> being    <a href="ideal_1.html#M2" target="_self">LeftLinearCombination</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E4:92_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font>
 <b>by </b><i/>;<br/><a NAME="E5:92_1"/>
<font color="Maroon">lc</font> is    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <font color="Maroon">F</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:92_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:92"/>
<font color="Maroon">F</font> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="ideal_1.html#K8" target="_self">-LeftIdeal</a> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:92_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 ;<br/><b>then consider </b><font color="Maroon">lc</font> being    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E3:92_2_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:92_2_1"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E4:92_2_1"/>
<font color="Maroon">lc</font> is    <a href="ideal_1.html#M3" target="_self">RightLinearCombination</a> of <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E5:92_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="ideal_1.html#K9" target="_self">-RightIdeal</a> 
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:92_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="ideal_1.html#K9" target="_self">-RightIdeal</a> 
 ;<br/><b>then consider </b><font color="Maroon">lc</font> being    <a href="ideal_1.html#M3" target="_self">RightLinearCombination</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E4:92_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font>
 <b>by </b><i/>;<br/><a NAME="E5:92_2"/>
<font color="Maroon">lc</font> is    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <font color="Maroon">F</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:92_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:92"/>
<font color="Maroon">F</font> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="ideal_1.html#K9" target="_self">-RightIdeal</a> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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