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

<b>assume </b><a NAME="E1:69"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">m</font> <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="newton.html#K9">choose</a> <font color="Maroon">n</font>
 ;<br/>

<b>assume </b><a NAME="E2:69"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3
 ;<br/>

<a NAME="E3:69"/><b>then </b><i><font color="Green">E40</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/>
<a NAME="E4:69"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <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>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:69"/><b>then </b>
<font color="Maroon">m</font> <a href="binop_2.html#K24">*</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#NR2" target="_self">&gt;=</a> <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>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E6:69"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,  <a href="newton.html#NM1">Prime</a>] <b>means </b>( <font color="Maroon">a<sub>3</sub></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">a<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 );<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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon">f1</font> =  <a href="finseq_1.html#K14">Sgm</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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> ;<br/>
<a NAME="E7:69"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a><br/> for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">m</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R1">=</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> <font color="Olive">m</font></span>)</span></span>)</span> where <font color="Olive">p'</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">m</font>,<font color="Olive">p'</font>] </span>}</span>  holds <br/> <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Olive">X</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>from </b><i><a class="ref" href="nat_4.html#S1" target="_self">NAT_4:sch 1</a>();<br/></i>
<a NAME="E8:69"/><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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,  <a href="newton.html#NM1">Prime</a>] <b>means </b>(  <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">a<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a<sub>3</sub></font> &amp; <font color="Maroon">a<sub>3</sub></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">a<sub>1</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> 3 &amp; <font color="Maroon">a<sub>3</sub></font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 );<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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon">f2</font> =  <a href="finseq_1.html#K14">Sgm</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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> ;<br/>
<a NAME="E9:69"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a><br/> for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">m</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R1">=</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> <font color="Olive">m</font></span>)</span></span>)</span> where <font color="Olive">p'</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Olive">m</font>,<font color="Olive">p'</font>] </span>}</span>  holds <br/> <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Olive">X</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>from </b><i><a class="ref" href="nat_4.html#S1" target="_self">NAT_4:sch 1</a>();<br/></i>
<a NAME="E10:69"/><b>then </b><i><font color="Green">E54</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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,  <a href="newton.html#NM1">Prime</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a<sub>3</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 );<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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon">f3</font> =  <a href="finseq_1.html#K14">Sgm</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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> ;<br/>
<a NAME="E11:69"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a><br/> for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">m</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R1">=</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> <font color="Olive">m</font></span>)</span></span>)</span> where <font color="Olive">p'</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>,<font color="Olive">m</font>,<font color="Olive">p'</font>] </span>}</span>  holds <br/> <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Olive">X</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>from </b><i><a class="ref" href="nat_4.html#S1" target="_self">NAT_4:sch 1</a>();<br/></i>
<a NAME="E12:69"/><b>then </b><i><font color="Green">E56</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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">k</font> = <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">n1</font> =  <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span>;<br/>
<b>set </b><font color="Maroon">n2</font> =  <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span>;<br/>
<b>set </b><font color="Maroon">n3</font> =  <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span>;<br/>
<a NAME="E13:69"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="txt" href="nat_4.html#E22:69"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E14:69"/><b>then </b><i><font color="Green">E58</font></i>: 
<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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E23:69"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E15:69"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a><br/> for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">m</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R1">=</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> <font color="Olive">m</font></span>)</span></span>)</span> where <font color="Olive">p'</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">m</font>,<font color="Olive">p'</font>] </span>}</span>  &amp; <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> <font color="Olive">m</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> holds <br/><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Olive">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Olive">m</font>
 
<b>from </b><i><a class="ref" href="nat_4.html#S3" target="_self">NAT_4:sch 3</a>();<br/></i>
<a NAME="E16:69"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a><br/> for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">m</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R1">=</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> <font color="Olive">m</font></span>)</span></span>)</span> where <font color="Olive">p'</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Olive">m</font>,<font color="Olive">p'</font>] </span>}</span>  &amp; <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> <font color="Olive">m</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> holds <br/><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Olive">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Olive">m</font>
 
<b>from </b><i><a class="ref" href="nat_4.html#S3" target="_self">NAT_4:sch 3</a>();<br/></i>
<a NAME="E17:69"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a><br/> for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">m</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R1">=</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> <font color="Olive">m</font></span>)</span></span>)</span> where <font color="Olive">p'</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>,<font color="Olive">m</font>,<font color="Olive">p'</font>] </span>}</span>  &amp; <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> <font color="Olive">m</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> holds <br/><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Olive">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Olive">m</font>
 
<b>from </b><i><a class="ref" href="nat_4.html#S3" target="_self">NAT_4:sch 3</a>();<br/></i>
<a NAME="E18:69"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a><br/> for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">m</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R1">=</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> <font color="Olive">m</font></span>)</span></span>)</span> where <font color="Olive">p'</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">m</font>,<font color="Olive">p'</font>] </span>}</span>  &amp; not <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> <font color="Olive">m</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> holds <br/><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Olive">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>from </b><i><a class="ref" href="nat_4.html#S2" target="_self">NAT_4:sch 2</a>();<br/></i>
<a NAME="E19:69"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a><br/> for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">m</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R1">=</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> <font color="Olive">m</font></span>)</span></span>)</span> where <font color="Olive">p'</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Olive">m</font>,<font color="Olive">p'</font>] </span>}</span>  &amp; not <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> <font color="Olive">m</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> holds <br/><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Olive">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>from </b><i><a class="ref" href="nat_4.html#S2" target="_self">NAT_4:sch 2</a>();<br/></i>
<a NAME="E20:69"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a><br/> for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">m</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R1">=</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> <font color="Olive">m</font></span>)</span></span>)</span> where <font color="Olive">p'</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>,<font color="Olive">m</font>,<font color="Olive">p'</font>] </span>}</span>  &amp; not <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> <font color="Olive">m</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> holds <br/><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Olive">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>from </b><i><a class="ref" href="nat_4.html#S2" target="_self">NAT_4:sch 2</a>();<br/></i>
<div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">n</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>
 ;<br/><a NAME="E2:69_1"/><i><font color="Green">E70</font></i>: 
 <a href="square_1.html#K7">sqrt</a> 2 <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K7">sqrt</a> 3
 <b>by </b><i><a class="ref" href="square_1.html#T95">SQUARE_1:95</a></i>;<br/><a NAME="E3:69_1"/>
 <a href="square_1.html#K7">sqrt</a> 3 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="square_1.html#T94">SQUARE_1:94</a>, <a class="txt" href="nat_4.html#E20:69"><i><font color="Green">E66</font></i></a></i>;<br/><a NAME="E4:69_1"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="square_1.html#K7">sqrt</a> 2 <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K7">sqrt</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="E5:69_1"/>
 <a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="nat_4.html#E20:69"><i><font color="Green">E66</font></i></a>, <a class="ref" href="square_1.html#T93">SQUARE_1:93</a></i>;<br/><a NAME="E6:69_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E7:69_1"/><b>then </b>
 <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> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="square_1.html#T97">SQUARE_1:97</a></i>;<br/><a NAME="E8:69_1"/><b>then </b>
 <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>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <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#T97">SQUARE_1:97</a></i>;<br/><a NAME="E9:69_1"/><b>then </b>
 <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>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="pepin.html#K2">^2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>;<br/><b>hence </b><a NAME="E10:69_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E26:69"><i><font color="Green">E83</font></i></a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E72</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_2"/><i><font color="Green">E73</font></i>: 
<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="binop_2.html#K12">/</a> 3
 ;<br/><a NAME="E2:69_2"/>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 3 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E20:69"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:69_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> 3 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<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
 <b>by </b><i><a class="ref" href="xreal_1.html#T76">XREAL_1:76</a></i>;<br/><b>hence </b><a NAME="E4:69_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E23:69"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a>  st <font color="Olive">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> holds <br/>( <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>

<b>assume </b><a NAME="E1:69_3"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:69_3"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E12:69"><i><font color="Green">E56</font></i></a>, <a class="txt" href="nat_4.html#E20:69"><i><font color="Green">E66</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_3_1"/>
<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> <font color="Maroon">m</font></span>)</span> <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> 
 ;<br/><a NAME="E2:69_3_1"/><b>then </b>
 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> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <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> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p'</font>] )
 ;<br/><b>hence </b><a NAME="E3:69_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E27:69"><i><font color="Green">E86</font></i></a>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:69_3"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_3_2"/>
<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> <font color="Maroon">m</font></span>)</span> <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> 
 ;<br/><a NAME="E2:69_3_2"/><b>then </b>
 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> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <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> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p'</font>] )
 ;<br/><b>hence </b><a NAME="E3:69_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E27:69"><i><font color="Green">E86</font></i></a>, , <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:69_3"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_4.html#E25:69"><i><font color="Green">E80</font></i></a>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_3_3"/>
<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> <font color="Maroon">m</font></span>)</span> <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> 
 ;<br/><a NAME="E2:69_3_3"/><b>then </b>
 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> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <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> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p'</font>] )
 ;<br/><b>hence </b><a NAME="E3:69_3_3"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:69_3"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:69"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a>  st <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> holds <br/>( <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>

<b>assume </b><a NAME="E1:69_4"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">n</font> <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> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_4_1"/>
<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> <font color="Maroon">m</font></span>)</span> <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> 
 ;<br/><a NAME="E2:69_4_1"/><b>then </b>
 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> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <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> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p'</font>] )
 ;<br/><a NAME="E3:69_4_1"/><b>then </b>
<font color="Maroon">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>
 <b>by </b><i>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/><b>hence </b><a NAME="E4:69_4_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="nat_4.html#E27:69"><i><font color="Green">E86</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:69_4"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_4_2"/>
<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> <font color="Maroon">m</font></span>)</span> <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> 
 ;<br/><a NAME="E2:69_4_2"/><b>then </b>
 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> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <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> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p'</font>] )
 ;<br/><a NAME="E3:69_4_2"/><b>then </b>
<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#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/><b>hence </b><a NAME="E4:69_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E26:69"><i><font color="Green">E83</font></i></a>, <a class="txt" href="nat_4.html#E27:69"><i><font color="Green">E86</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:69_4"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_4.html#E25:69"><i><font color="Green">E80</font></i></a>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_4_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_4_3"/>
( <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_4_3_1"><b>suppose </b></a><a NAME="E1:69_4_3_1"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:69_4_3_1"/><b>then </b>
<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> <font color="Maroon">m</font></span>)</span> <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="nat_4.html#E27:69"><i><font color="Green">E86</font></i></a></i>;<br/><b>hence </b><a NAME="E3:69_4_3_1"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:69_4"><i><font color="Green">E78</font></i></a>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_4_3_2"><b>suppose </b></a><a NAME="E1:69_4_3_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_4_3_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_4_3_2_1"/>
<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> <font color="Maroon">m</font></span>)</span> <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> 
 ;<br/><a NAME="E2:69_4_3_2_1"/><b>then </b>
 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> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <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> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p'</font>] )
 ;<br/><b>hence </b><a NAME="E3:69_4_3_2_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:69_4_3_2"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E25:69"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a>  st <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> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/>( <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>

<b>assume </b><a NAME="E1:69_5"/><i><font color="Green">E81</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="binop_2.html#K12">/</a> 3 <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> <font color="Maroon">n</font> )
 ;<br/>

<a NAME="E2:69_5"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E12:69"><i><font color="Green">E56</font></i></a>, <a class="txt" href="nat_4.html#E20:69"><i><font color="Green">E66</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_5_1"/>
<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> <font color="Maroon">m</font></span>)</span> <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> 
 ;<br/><a NAME="E2:69_5_1"/><b>then </b>
 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> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <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> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p'</font>] )
 ;<br/><b>hence </b><a NAME="E3:69_5_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:69_5"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_5_2"/>
<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> <font color="Maroon">m</font></span>)</span> <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> 
 ;<br/><a NAME="E2:69_5_2"/><b>then </b>
 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> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <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> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p'</font>] )
 ;<br/><b>hence </b><a NAME="E3:69_5_2"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:69_5"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_4.html#E25:69"><i><font color="Green">E80</font></i></a>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_5_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_5_3"/>
<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> <font color="Maroon">m</font></span>)</span> <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> 
 ;<br/><a NAME="E2:69_5_3"/><b>then </b>
 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> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <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> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p'</font>] )
 ;<br/><b>hence </b><a NAME="E3:69_5_3"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:69_5"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_4.html#E1:69_5"><i><font color="Green">E81</font></i></a>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:69"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a>  st  <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 holds <br/>( <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>

<b>assume </b><a NAME="E1:69_6"/><i><font color="Green">E84</font></i>: 
(  <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 )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_6_1"/>
<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> <font color="Maroon">m</font></span>)</span> <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> 
 ;<br/><a NAME="E2:69_6_1"/><b>then </b>
 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> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <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> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p'</font>] )
 ;<br/><b>hence </b><a NAME="E3:69_6_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:69_6"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_6_2"/>
<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> <font color="Maroon">m</font></span>)</span> <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> 
 ;<br/><a NAME="E2:69_6_2"/><b>then </b>
 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> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <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> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p'</font>] )
 ;<br/><a NAME="E3:69_6_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/><b>hence </b><a NAME="E4:69_6_2"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E26:69"><i><font color="Green">E83</font></i></a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:69_6"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_6_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_6_3"/>
( <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_6_3_1"><b>suppose </b></a><a NAME="E1:69_6_3_1"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:69_6_3_1"/><b>then </b>
<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> <font color="Maroon">m</font></span>)</span> <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:69_6_3_1"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E24:69"><i><font color="Green">E77</font></i></a>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_6_3_2"><b>suppose </b></a><a NAME="E1:69_6_3_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_6_3_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_6_3_2_1"/>
<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> <font color="Maroon">m</font></span>)</span> <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> 
 ;<br/><a NAME="E2:69_6_3_2_1"/><b>then </b>
 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> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <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> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p'</font>] )
 ;<br/><b>hence </b><a NAME="E3:69_6_3_2_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:69_6_3_2"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E25:69"><i><font color="Green">E80</font></i></a>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E27:69"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a>  st <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> holds <br/>( <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>

<b>assume </b><a NAME="E1:69_7"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">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>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_7_1"/>
<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> <font color="Maroon">m</font></span>)</span> <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> 
 ;<br/><a NAME="E2:69_7_1"/><b>then </b>
 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> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <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> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p'</font>] )
 ;<br/><b>hence </b><a NAME="E3:69_7_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:69_7"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_4.html#E25:69"><i><font color="Green">E80</font></i></a>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_7_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_7_2"/>
<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> <font color="Maroon">m</font></span>)</span> <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> 
 ;<br/><a NAME="E2:69_7_2"/><b>then </b>
 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> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <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> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p'</font>] )
 ;<br/><a NAME="E3:69_7_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/><b>hence </b><a NAME="E4:69_7_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:69_7"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_7_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_7_3"/>
( <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_7_3_1"><b>suppose </b></a><a NAME="E1:69_7_3_1"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:69_7_3_1"/><b>then </b>
<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> <font color="Maroon">m</font></span>)</span> <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:69_7_3_1"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_7_3_2"><b>suppose </b></a><a NAME="E1:69_7_3_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_7_3_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_7_3_2_1"/>
<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> <font color="Maroon">m</font></span>)</span> <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> 
 ;<br/><a NAME="E2:69_7_3_2_1"/><b>then </b>
 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> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <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> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p'</font>] )
 ;<br/><b>hence </b><a NAME="E3:69_7_3_2_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:69_7_3_2"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E28:69"/>
for <font color="Olive">p</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> is <a href="int_2.html#V1">prime</a> holds <br/><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <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"><span class="p3">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p5">{<span class="default"> <span class="p0">(<span class="default"><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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p5">{<span class="default"> <span class="p0">(<span class="default"><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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_8"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:69_8"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">p</font> is <a href="int_2.html#V1">prime</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">n1</font> =  <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">n2</font> =  <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">n3</font> =  <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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a></i>;<br/>
<i><font color="Green">E91</font></i>: <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"><span class="p3">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p5">{<span class="default"> <span class="p0">(<span class="default"><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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p5">{<span class="default"> <span class="p0">(<span class="default"><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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span></span>)</span> = 
<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"><font color="Maroon">n1</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">n2</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n3</font></span>)</span>
<b>by </b><i>, <a class="ref" href="nat_3.html#T28">NAT_3:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n2</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n3</font></span>)</span>
<b>by </b><i>, <a class="ref" href="nat_3.html#T28">NAT_3:28</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_8_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_8_2"/>
( <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> or ( <font color="Maroon">n</font> <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> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> ) or ( <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> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) or (  <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 ) or <font color="Maroon">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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_8_2_1"><b>suppose </b></a><a NAME="E1:69_8_2_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:69_8_2_1"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="txt" href="nat_4.html#E20:69"><i><font color="Green">E66</font></i></a>, , <a class="txt" href="nat_4.html#E12:69"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E3:69_8_2_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="txt" href="nat_4.html#E5:69_8"><i><font color="Green">E91</font></i></a></i>;<br/><a NAME="E4:69_8_2_1"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="txt" href="nat_4.html#E5:69_8"><i><font color="Green">E91</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_8_2_1"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <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"><span class="p3">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p5">{<span class="default"> <span class="p0">(<span class="default"><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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p5">{<span class="default"> <span class="p0">(<span class="default"><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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:69_8"><i><font color="Green">E90</font></i></a>, <a class="txt" href="nat_4.html#E1:69_8_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="nat_4.html#E2:69_8_2_1"><i><font color="Green">E93</font></i></a>, , , <a class="txt" href="nat_4.html#E5:69_8"><i><font color="Green">E91</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_8_2_2"><b>suppose </b></a><a NAME="E1:69_8_2_2"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">n</font> <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> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><a NAME="E2:69_8_2_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="txt" href="nat_4.html#E5:69_8"><i><font color="Green">E91</font></i></a></i>;<br/><a NAME="E3:69_8_2_2"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="txt" href="nat_4.html#E5:69_8"><i><font color="Green">E91</font></i></a></i>;<br/><b>hence </b><a NAME="E4:69_8_2_2"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <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"><span class="p3">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p5">{<span class="default"> <span class="p0">(<span class="default"><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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p5">{<span class="default"> <span class="p0">(<span class="default"><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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:69_8"><i><font color="Green">E90</font></i></a>, , , , <a class="txt" href="nat_4.html#E5:69_8"><i><font color="Green">E91</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_8_2_3"><b>suppose </b></a><a NAME="E1:69_8_2_3"/><i><font color="Green">E98</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="binop_2.html#K12">/</a> 3 <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> <font color="Maroon">n</font> )
 ;<br/><div class="add"><a NAME="E2:69_8_2_3"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="txt" href="nat_4.html#E20:69"><i><font color="Green">E66</font></i></a>, , <a class="txt" href="nat_4.html#E12:69"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E3:69_8_2_3"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_4.html#E27:69"><i><font color="Green">E86</font></i></a>, , <a class="txt" href="nat_4.html#E5:69_8"><i><font color="Green">E91</font></i></a></i>;<br/><a NAME="E4:69_8_2_3"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_4.html#E27:69"><i><font color="Green">E86</font></i></a>, , <a class="txt" href="nat_4.html#E5:69_8"><i><font color="Green">E91</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_8_2_3"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <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"><span class="p3">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p5">{<span class="default"> <span class="p0">(<span class="default"><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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p5">{<span class="default"> <span class="p0">(<span class="default"><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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:69_8"><i><font color="Green">E90</font></i></a>, , , <a class="txt" href="nat_4.html#E27:69"><i><font color="Green">E86</font></i></a>, , <a class="txt" href="nat_4.html#E5:69_8"><i><font color="Green">E91</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_8_2_4"><b>suppose </b></a><a NAME="E1:69_8_2_4"/><i><font color="Green">E101</font></i>: 
(  <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 )
 ;<br/><div class="add"><a NAME="E2:69_8_2_4"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="txt" href="nat_4.html#E5:69_8"><i><font color="Green">E91</font></i></a></i>;<br/><a NAME="E3:69_8_2_4"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="txt" href="nat_4.html#E5:69_8"><i><font color="Green">E91</font></i></a></i>;<br/><b>hence </b><a NAME="E4:69_8_2_4"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <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"><span class="p3">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p5">{<span class="default"> <span class="p0">(<span class="default"><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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p5">{<span class="default"> <span class="p0">(<span class="default"><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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:69_8"><i><font color="Green">E90</font></i></a>, , , , <a class="txt" href="nat_4.html#E5:69_8"><i><font color="Green">E91</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_8_2_5"><b>suppose </b></a><a NAME="E1:69_8_2_5"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">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>
 ;<br/><div class="add"><a NAME="E2:69_8_2_5"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="txt" href="nat_4.html#E5:69_8"><i><font color="Green">E91</font></i></a></i>;<br/><a NAME="E3:69_8_2_5"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="txt" href="nat_4.html#E5:69_8"><i><font color="Green">E91</font></i></a></i>;<br/><b>hence </b><a NAME="E4:69_8_2_5"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <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"><span class="p3">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p5">{<span class="default"> <span class="p0">(<span class="default"><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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p5">{<span class="default"> <span class="p0">(<span class="default"><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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </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> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p</font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:69_8"><i><font color="Green">E90</font></i></a>, , , , <a class="txt" href="nat_4.html#E5:69_8"><i><font color="Green">E91</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E29:69"/>
<font color="Maroon">m</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> <font color="Maroon">m</font></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> <font color="Maroon">m</font> <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> <font color="Maroon">m</font></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> <font color="Maroon">m</font> <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> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></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> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="nat_4.html#E21:69"><i><font color="Green">E68</font></i></a>, </i>;<br/>


</div>
