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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:72"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:72_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 4000 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4000 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1"><b>suppose </b></a><a NAME="E1:72_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 4000
 ;<br/><div class="add"><a NAME="E2:72_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 4001
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E3:72_1_1"/>
 ex <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_2"><b>suppose </b></a><a NAME="E1:72_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4000
 ;<br/><div class="add"><b>assume </b><a NAME="E2:72_1_2"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a>  holds <br/>(  not <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> or  not <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> )
 ;<br/><b>set </b><font color="Maroon">m</font> = <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font>;<br/><a NAME="E3:72_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E4:72_1_2"/><b>then </b><i><font color="Green">E47</font></i>: 
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 <a href="binop_2.html#K24">*</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>set </b><font color="Maroon">X1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> ;<br/><b>set </b><font color="Maroon">X2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : (  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> ;<br/><b>set </b><font color="Maroon">X3</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> ;<br/><a NAME="E5:72_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E6:72_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 </span>}</span> ;<br/><b>set </b><font color="Maroon">mm</font> = <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span></span><a href="int_1.html#K2">\]</a></span>;<br/><b>reconsider </b><font color="Maroon">mm</font> = <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span></span><a href="int_1.html#K2">\]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T80">INT_1:80</a></i>;<br/><b>set </b><font color="Maroon">XX</font> =  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">mm</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_2_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/><b>assume </b><a NAME="E1:72_1_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 </span>}</span> 
 ;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="newton.html#NM1">Prime</a><b> such that </b><br/><a NAME="E3:72_1_2_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 )
 <b>by </b><i><a class="txt" href="nat_4.html#E21:72_1_2"><i><font color="Green">E72</font></i></a></i>;<br/><a NAME="E4:72_1_2_1"/>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span></span><a href="int_1.html#K2">\]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/><a NAME="E5:72_1_2_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span></span><a href="int_1.html#K2">\]</a></span>
 <b>by </b><i><a class="txt" href="nat_4.html#E23:72_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E6:72_1_2_1"/><i><font color="Green">E56</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>thus </b><a NAME="E8:72_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">mm</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E23:72_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="nat_4.html#E24:72_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="nat_4.html#E25:72_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/></div><b>end;</b></div><a NAME="E9:72_1_2"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">mm</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E10:72_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
( <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 </span>}</span>  is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><div><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_1_2_2"/><i><font color="Green">E61</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 </span>}</span> 
 ;<br/><a NAME="E2:72_1_2_2"/>
 ex <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 )
 <b>by </b><i><a class="txt" href="nat_4.html#E24:72_1_2"><i><font color="Green">E74</font></i></a></i>;<br/><b>hence </b><a NAME="E3:72_1_2_2"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:72_1_2_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : (  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> 
 ;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="newton.html#NM1">Prime</a><b> such that </b><br/><a NAME="E3:72_1_2_3"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 &amp; <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="nat_4.html#E24:72_1_2"><i><font color="Green">E74</font></i></a></i>;<br/><a NAME="E4:72_1_2_3"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, <a class="txt" href="nat_4.html#E25:72_1_2"><i><font color="Green">E75</font></i></a>, </i>;<br/><a NAME="E5:72_1_2_3"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:72_1_2_3"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E25:72_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="nat_1.html#T71">NAT_1:71</a></i>;<br/><a NAME="E7:72_1_2_3"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E25:72_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="newton.html#T10">NEWTON:10</a></i>;<br/><b>hence </b><a NAME="E8:72_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 </span>}</span> 
 <b>by </b><i><a class="txt" href="nat_4.html#E25:72_1_2"><i><font color="Green">E75</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E13:72_1_2"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : (  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E14:72_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : (  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 </span>}</span> </span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E21:72_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="nat_4.html#E23:72_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="nat_4.html#E6:72_1_2"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E15:72_1_2"/>
<font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 3 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 <a href="binop_2.html#K12">/</a> 3
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T74">XREAL_1:74</a></i>;<br/><a NAME="E16:72_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 3</span>)</span> <a href="binop_2.html#K11">*</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="binop_2.html#K24">*</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E17:72_1_2"/><b>then </b>
 <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 </span>}</span> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E18:72_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : (  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E24:72_1_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_1_2_4"/><i><font color="Green">E69</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:72_1_2_4"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="nat_4.html#E27:72_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:72_1_2_4"/>
 ex <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="nat_4.html#E42:72_1_2"><i><font color="Green">E78</font></i></a></i>;<br/><b>hence </b><a NAME="E5:72_1_2_4"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E19:72_1_2"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E20:72_1_2"/><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> </span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : (  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E26:72_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="rvsum_1.html#T124">RVSUM_1:124</a>, <a class="ref" href="finseq_3.html#T49">FINSEQ_3:49</a>, , <a class="txt" href="nat_4.html#E14:72_1_2"><i><font color="Green">E64</font></i></a></i>;<br/><b>set </b><font color="Maroon">XX</font> =  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font></span>)</span>;<br/><a NAME="E21:72_1_2"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">4 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E27:72_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xreal_1.html#T68">XREAL_1:68</a>, <a class="txt" href="nat_4.html#E25:72_1_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="nat_4.html#E20:72_1_2"><i><font color="Green">E71</font></i></a></i>;<br/><a NAME="E22:72_1_2"/>
<span class="p1">(<span class="default">4 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E23:72_1_2"/><b>then </b><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default">4 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">4 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E42:72_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E24:72_1_2"/><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E25:72_1_2"/><i><font color="Green">E75</font></i>: 
4 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T46">POWER:46</a></i>;<br/><a NAME="E26:72_1_2"/><i><font color="Green">E76</font></i>: 
4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E27:72_1_2"/><i><font color="Green">E77</font></i>: 
4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 3</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E28:72_1_2"/>
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0 <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E29:72_1_2"/><b>then </b>
4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T44">POWER:44</a></i>;<br/><a NAME="E30:72_1_2"/><b>then </b>
<span class="p1">(<span class="default">4 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">4 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E46:72_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E31:72_1_2"/><b>then </b>
<span class="p1">(<span class="default">4 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">4 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E44:72_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E32:72_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default">4 <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E33:72_1_2"/><b>then </b>
4 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">4 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E34:72_1_2"/><b>then </b>
<span class="p1">(<span class="default">4 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">4 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default">4 <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">4 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E55:72_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="nat_4.html#E49:72_1_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xreal_1.html#T74">XREAL_1:74</a></i>;<br/><a NAME="E35:72_1_2"/><b>then </b>
<span class="p1">(<span class="default">4 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">4 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E55:72_1_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><a NAME="E36:72_1_2"/><b>then </b>
4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T34">POWER:34</a></i>;<br/><a NAME="E37:72_1_2"/><b>then </b>
4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 3</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/><a NAME="E38:72_1_2"/><b>then </b>
4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 3</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="power.html#T32">POWER:32</a></i>;<br/><a NAME="E39:72_1_2"/><b>then </b>
( 4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 3</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> or 4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><a NAME="E40:72_1_2"/><b>then </b>
<span class="p1">(<span class="default">4 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 3</span>)</span></span>)</span> <a href="power.html#K4">to_power</a> 3 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> <a href="power.html#K4">to_power</a> 3
 <b>by </b><i><a class="txt" href="nat_4.html#E58:72_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="power.html#T42">POWER:42</a></i>;<br/><a NAME="E41:72_1_2"/><b>then </b>
4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> 3</span>)</span> <a href="binop_2.html#K11">*</a> 3</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> <a href="power.html#K4">to_power</a> 3
 <b>by </b><i><a class="ref" href="power.html#T38">POWER:38</a></i>;<br/><a NAME="E42:72_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
4 <a href="power.html#K4">to_power</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> 3</span>)</span>
 <b>by </b><i>, <a class="ref" href="power.html#T38">POWER:38</a></i>;<br/><b>set </b><font color="Maroon">n1</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>;<br/><a NAME="E43:72_1_2"/>
<span class="p1">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><a NAME="E44:72_1_2"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E45:72_1_2"/>
6 <a href="power.html#K2">-root</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 6 <a href="power.html#K2">-root</a> 0
 <b>by </b><i><a class="ref" href="power.html#T18">POWER:18</a>, </i>;<br/><a NAME="E46:72_1_2"/><b>then </b><i><font color="Green">E80</font></i>: 
6 <a href="power.html#K2">-root</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T6">POWER:6</a></i>;<br/><b>reconsider </b><font color="Maroon">n1</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T80">INT_1:80</a>, <a class="txt" href="nat_4.html#E71:72_1_2"><i><font color="Green">E86</font></i></a></i>;<br/><a NAME="E48:72_1_2"/>
<font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="heine.html#T7">HEINE:7</a></i>;<br/><a NAME="E49:72_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="power.html#K4">to_power</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="power.html#T46">POWER:46</a></i>;<br/><a NAME="E50:72_1_2"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> 6 <a href="power.html#K2">-root</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><a NAME="E51:72_1_2"/><b>then </b>
( <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> 6 <a href="power.html#K2">-root</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> or <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 6 <a href="power.html#K2">-root</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><a NAME="E52:72_1_2"/><b>then </b>
2 <a href="power.html#K4">to_power</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T44">POWER:44</a></i>;<br/><a NAME="E53:72_1_2"/><b>then </b>
<font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E78:72_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E54:72_1_2"/><b>then </b>
( <font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> or <font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> 2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><a NAME="E55:72_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="power.html#K4">to_power</a> 6 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> 6
 <b>by </b><i><a class="ref" href="power.html#T42">POWER:42</a></i>;<br/><a NAME="E56:72_1_2"/>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="square_1.html#T93">SQUARE_1:93</a></i>;<br/><a NAME="E57:72_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E58:72_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> 3 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K24">*</a> 3
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>reconsider </b><font color="Maroon">2n</font> = 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/><i><font color="Green">E84</font></i>: <span class="p1">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> 6 = 
<span class="p1">(<span class="default">6 <a href="power.html#K1">-root</a> <font color="Maroon">2n</font></span>)</span> <a href="newton.html#K2">|^</a> 6
<b>by </b><i><a class="ref" href="power.html#T47">POWER:47</a></i>
<br/>.= 
<font color="Maroon">2n</font>
<b>by </b><i><a class="ref" href="comptrig.html#T17">COMPTRIG:17</a></i>
;<br/><a NAME="E61:72_1_2"/>
<span class="p1">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> 6 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="power.html#K4">to_power</a> 6
 <b>by </b><i><a class="txt" href="nat_4.html#E71:72_1_2"><i><font color="Green">E86</font></i></a>, <a class="txt" href="nat_4.html#E68:72_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="power.html#T42">POWER:42</a></i>;<br/><a NAME="E62:72_1_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="wsierp_1.html#K2">|^</a> 6
 <b>by </b><i><a class="ref" href="power.html#T47">POWER:47</a>, <a class="txt" href="nat_4.html#E80:72_1_2"><i><font color="Green">E90</font></i></a></i>;<br/><a NAME="E63:72_1_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="power.html#K4">to_power</a> 6
 <b>by </b><i><a class="ref" href="power.html#T47">POWER:47</a></i>;<br/><a NAME="E64:72_1_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> 6
 <b>by </b><i><a class="txt" href="nat_4.html#E79:72_1_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E65:72_1_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> 6</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T38">POWER:38</a></i>;<br/><a NAME="E66:72_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> 3</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> 6</span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> 3</span>)</span>
 <b>by </b><i>, , <a class="ref" href="power.html#T42">POWER:42</a></i>;<br/><a NAME="E67:72_1_2"/><b>then </b>
4 <a href="power.html#K4">to_power</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> 6</span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> 3</span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E60:72_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E68:72_1_2"/><b>then </b><i><font color="Green">E85</font></i>: 
4 <a href="power.html#K4">to_power</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> 6</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> 3</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T38">POWER:38</a></i>;<br/><a NAME="E69:72_1_2"/>
81 <a href="hidden.html#R1">=</a> 9 <a href="binop_2.html#K24">*</a> 9
 ;<br/><a NAME="E70:72_1_2"/><b>then </b>
81 <a href="hidden.html#R1">=</a> 9 <a href="pepin.html#K2">^2</a> 
 <b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>;<br/><a NAME="E71:72_1_2"/><b>then </b><i><font color="Green">E86</font></i>: 
 <a href="square_1.html#K7">sqrt</a> 81 <a href="hidden.html#R1">=</a> 9
 <b>by </b><i><a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/><a NAME="E72:72_1_2"/>
42 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E73:72_1_2"/><b>then </b>
42 <a href="binop_2.html#K24">*</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E74:72_1_2"/><b>then </b>
81 <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E75:72_1_2"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> 81 <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="square_1.html#T95">SQUARE_1:95</a></i>;<br/><a NAME="E76:72_1_2"/><b>then </b>
9 <a href="binop_2.html#K24">*</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> 2
 <b>by </b><i><a class="txt" href="nat_4.html#E82:72_1_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E77:72_1_2"/><b>then </b>
18 <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default">18 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default">18 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E78:72_1_2"/><b>then </b><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default">18 <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default">18 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">20 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a>, <a class="txt" href="nat_4.html#E71:72_1_2"><i><font color="Green">E86</font></i></a></i>;<br/><a NAME="E79:72_1_2"/><i><font color="Green">E88</font></i>: 
2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E80:72_1_2"/><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K12">/</a> 3</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a>, </i>;<br/><a NAME="E81:72_1_2"/><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K12">/</a> 3</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a>, </i>;<br/><a NAME="E82:72_1_2"/><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 3</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a>, </i>;<br/>2 <a href="power.html#K4">to_power</a> 2 = 
2 <a href="wsierp_1.html#K2">|^</a> 2
<b>by </b><i><a class="ref" href="power.html#T46">POWER:46</a></i>
<br/>.= 
 <a href="rvsum_1.html#K16">Product</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">2,2</span><a href="finseq_1.html#K10">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T75">FINSEQ_2:75</a></i>
<br/>.= 
2 <a href="binop_2.html#K24">*</a> 2
<b>by </b><i><a class="ref" href="rvsum_1.html#T129">RVSUM_1:129</a></i>
;<br/><a NAME="E84:72_1_2"/><b>then </b>
4 <a href="power.html#K4">to_power</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T38">POWER:38</a></i>;<br/><a NAME="E85:72_1_2"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> 6</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> 3</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T65">POWER:65</a>, , <a class="txt" href="nat_4.html#E81:72_1_2"><i><font color="Green">E91</font></i></a></i>;<br/><a NAME="E86:72_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> 6</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> 3</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T63">POWER:63</a></i>;<br/><a NAME="E87:72_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> 6</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,2</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T63">POWER:63</a></i>;<br/><a NAME="E88:72_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K24">*</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> 6</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,2</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T60">POWER:60</a></i>;<br/><a NAME="E89:72_1_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> 6</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> 1
 <b>by </b><i><a class="ref" href="power.html#T60">POWER:60</a></i>;<br/><a NAME="E90:72_1_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 20 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E91:72_1_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 20 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="prepower.html#K5">-Root</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="prepower.html#T41">PREPOWER:41</a></i>;<br/><a NAME="E92:72_1_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 20 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="power.html#K2">-root</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#D1">POWER:def 1</a></i>;<br/><a NAME="E93:72_1_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 20 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default">6 <a href="power.html#K2">-root</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T52">POWER:52</a></i>;<br/><a NAME="E94:72_1_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 20 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 6</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T52">POWER:52</a></i>;<br/><a NAME="E95:72_1_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 20 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 6</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a>, </i>;<br/><a NAME="E96:72_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K12">/</a> 3</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">20 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K12">/</a> 3</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K12">/</a> 3</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T76">XREAL_1:76</a>, </i>;<br/><a NAME="E97:72_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K12">/</a> 3</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 20
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a>, </i>;<br/><a NAME="E98:72_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K12">/</a> 3</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 20
 <b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/><a NAME="E99:72_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K12">/</a> 3</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 20
 <b>by </b><i><a class="ref" href="power.html#T34">POWER:34</a>, </i>;<br/><a NAME="E100:72_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> 3</span>)</span></span>)</span> <a href="power.html#K4">to_power</a> 3 <a href="xxreal_0.html#R1">&lt;</a> 20 <a href="power.html#K4">to_power</a> 3
 <b>by </b><i>, <a class="ref" href="power.html#T42">POWER:42</a></i>;<br/><a NAME="E101:72_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> 3</span>)</span> <a href="binop_2.html#K11">*</a> 3</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 20 <a href="power.html#K4">to_power</a> 3
 <b>by </b><i><a class="ref" href="power.html#T38">POWER:38</a>, </i>;<br/><a NAME="E102:72_1_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 20 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/><a NAME="E103:72_1_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">20 <a href="power.html#K4">to_power</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">20 <a href="power.html#K4">to_power</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>;<br/><a NAME="E104:72_1_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">20 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> 20
 <b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/><a NAME="E105:72_1_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">20 <a href="power.html#K4">to_power</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default">20 <a href="power.html#K4">to_power</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> 20
 <b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>;<br/><a NAME="E106:72_1_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">20 <a href="power.html#K4">to_power</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> 20</span>)</span> <a href="binop_2.html#K11">*</a> 20
 <b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/><a NAME="E107:72_1_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">20 <a href="binop_2.html#K24">*</a> 20</span>)</span> <a href="binop_2.html#K24">*</a> 20
 <b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/><a NAME="E108:72_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> 8000 <a href="binop_2.html#K12">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T76">XREAL_1:76</a></i>;<br/><b>hence </b><a NAME="E109:72_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
