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<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">I</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> holds <br/> ( <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ideal_1.html#K17" title="IDEAL_1:func.17">sqrt</a> <font color="Olive" title="b1">I</font> iff  ex <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Olive" title="b2">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ideal_1.html#K17" title="IDEAL_1:func.17">sqrt</a> <font color="Olive" title="b1">I</font> )</span><br/><b>let </b><font color="Maroon" title="c2">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> holds <br/> ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ideal_1.html#K17" title="IDEAL_1:func.17">sqrt</a> <font color="Maroon" title="c2">I</font> iff  ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Olive" title="b1">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ideal_1.html#K17" title="IDEAL_1:func.17">sqrt</a> <font color="Maroon" title="c2">I</font> )</span><br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ideal_1.html#K17" title="IDEAL_1:func.17">sqrt</a> <font color="Maroon" title="c2">I</font> iff  ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c3">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ideal_1.html#K17" title="IDEAL_1:func.17">sqrt</a> <font color="Maroon" title="c2">I</font> )</span><br/>





<div><i><font color="Green" title="E59">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="150_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:150_1"/><i><font color="Green" title="E60">A2</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ideal_1.html#K17" title="IDEAL_1:func.17">sqrt</a> <font color="Maroon" title="c2">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c3">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ideal_1.html#K17" title="IDEAL_1:func.17">sqrt</a> <font color="Maroon" title="c2">I</font></span><br/><a NAME="E2:150_1"/>
<font color="Maroon" title="c3">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="ref" href="binom.html#T8" title="BINOM:th.8">BINOM:8</a></i>;<br/><b>hence </b><a NAME="E3:150_1"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c3">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ideal_1.html#K17" title="IDEAL_1:func.17">sqrt</a> <font color="Maroon" title="c2">I</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:150_1"><i><font color="Green" title="E60">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:150_2"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c3">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ideal_1.html#K17" title="IDEAL_1:func.17">sqrt</a> <font color="Maroon" title="c2">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ideal_1.html#K17" title="IDEAL_1:func.17">sqrt</a> <font color="Maroon" title="c2">I</font></span><br/><b>then consider </b><font color="Maroon" title="c4">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:150_2"/><i><font color="Green" title="E60">A3</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ideal_1.html#K17" title="IDEAL_1:func.17">sqrt</a> <font color="Maroon" title="c2">I</font>
 ;<br/><b>consider </b><font color="Maroon" title="c5">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E5:150_2"/><i><font color="Green" title="E61">A4</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> &amp;  ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c5">d</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> )
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:150_2"><i><font color="Green" title="E60">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c6">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E7:150_2"/><i><font color="Green" title="E62">A5</font></i>: 
<font color="Maroon" title="c5">d</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E5:150_2"><i><font color="Green" title="E61">A4</font></i></a></i>;<br/><a NAME="E8:150_2"/>
<font color="Maroon" title="c3">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E5:150_2"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="binom.html#T12" title="BINOM:th.12">BINOM:12</a></i>;<br/><b>hence </b><a NAME="E9:150_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ideal_1.html#K17" title="IDEAL_1:func.17">sqrt</a> <font color="Maroon" title="c2">I</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E7:150_2"><i><font color="Green" title="E62">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:150"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ideal_1.html#K17" title="IDEAL_1:func.17">sqrt</a> <font color="Maroon" title="c2">I</font> iff  ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c3">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ideal_1.html#K17" title="IDEAL_1:func.17">sqrt</a> <font color="Maroon" title="c2">I</font> )
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:150"><i><font color="Green" title="E59">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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