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<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</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="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</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="151_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:151_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="ideal_1.html#K17" target="_self">sqrt</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:151_1"/>
<font color="Maroon">a</font> <a href="binom.html#K2">|^</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="binom.html#T8">BINOM:8</a></i>;<br/><b>hence </b><a NAME="E3:151_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a</font> <a href="binom.html#K2">|^</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="ideal_1.html#K17" target="_self">sqrt</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="151_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:151_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a</font> <a href="binom.html#K2">|^</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="ideal_1.html#K17" target="_self">sqrt</a> <font color="Maroon">I</font>
 ;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:151_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> <a href="binom.html#K2">|^</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="ideal_1.html#K17" target="_self">sqrt</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E4:151_2"/>
<font color="Maroon">a</font> <a href="binom.html#K2">|^</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> :  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">d</font> <a href="binom.html#K2">|^</a> <font color="Olive">m</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#T13" target="_self">Th13</a></i>;<br/><b>then consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E6:151_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">a</font> <a href="binom.html#K2">|^</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp;  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">d</font> <a href="binom.html#K2">|^</a> <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 ;<br/><b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:151_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">d</font> <a href="binom.html#K2">|^</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E9:151_2"/>
<font color="Maroon">a</font> <a href="binom.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="binom.html#K2">|^</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T14" target="_self">Th14</a>, <a class="ref" href="binom.html#T12">BINOM:12</a></i>;<br/><a NAME="E10:151_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> :  ex <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">g</font> <a href="binom.html#K2">|^</a> <font color="Olive">l</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#T17" target="_self">Th17</a></i>;<br/><b>hence </b><a NAME="E11:151_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="ideal_1.html#K17" target="_self">sqrt</a> <font color="Maroon">I</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:151"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="ideal_1.html#K17" target="_self">sqrt</a> <font color="Maroon">I</font> iff  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a</font> <a href="binom.html#K2">|^</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="ideal_1.html#K17" target="_self">sqrt</a> <font color="Maroon">I</font> )
 <b>by </b><i/>;<br/>


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