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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<div><i><font color="Green">E60</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">E61</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ideal_1.html#K17" target="_self">sqrt</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:151_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="binom.html#K2">|^</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></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">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>3</sub></font> <a href="binom.html#K2">|^</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ideal_1.html#K17" target="_self">sqrt</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:151_1"><i><font color="Green">E61</font></i></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">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>3</sub></font> <a href="binom.html#K2">|^</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ideal_1.html#K17" target="_self">sqrt</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:151_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="binom.html#K2">|^</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ideal_1.html#K17" target="_self">sqrt</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E4:151_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="binom.html#K2">|^</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="binom.html#K2">|^</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:151_2"><i><font color="Green">E61</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:151_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="binom.html#K2">|^</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>5</sub></font> <a href="binom.html#K2">|^</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:151_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="binom.html#K2">|^</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E6:151_2"><i><font color="Green">E62</font></i></a></i>;<br/><a NAME="E9:151_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="binom.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="binom.html#K2">|^</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E6:151_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="binom.html#T12">BINOM:12</a></i>;<br/><a NAME="E10:151_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="binom.html#K2">|^</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E8:151_2"><i><font color="Green">E63</font></i></a></i>;<br/><b>hence </b><a NAME="E11:151_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ideal_1.html#K17" target="_self">sqrt</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:151"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ideal_1.html#K17" target="_self">sqrt</a> <font color="Maroon">c<sub>2</sub></font> iff ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>3</sub></font> <a href="binom.html#K2">|^</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ideal_1.html#K17" target="_self">sqrt</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:151"><i><font color="Green">E60</font></i></a></i>;<br/>


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