<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:53"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> ;<br/>
<a NAME="E2:53"/><i><font color="Green">E53</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<a NAME="E1:53_1"/><i><font color="Green">E54</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p2"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2">\]</a></span></span>)</span> <a href="binop_2.html#K23">+</a> 1;<br/>
<b>take </b>
<span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p2"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2">\]</a></span></span>)</span> <a href="binop_2.html#K23">+</a> 1
;<br/>

<a NAME="E1:53_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2">\]</a></span>
 
<b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/>
<a NAME="E2:53_1_1"/><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="int_2.html#K1">abs</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2">\]</a></span>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/>
<a NAME="E3:53_1_1"/>
 <a href="int_2.html#K1">abs</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p2"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2">\]</a></span></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:53_1_1"/><b>then </b>
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p2"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2">\]</a></span></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E2:53_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E5:53_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p2"><a href="int_1.html#K2">[/</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K2">\]</a></span></span>)</span> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E1:53_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:53_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:53_1"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> ) </span>}</span> ;<br/>
<a NAME="E4:53_1"/><i><font color="Green">E56</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="newton.html#T91">NEWTON:91</a></i>;<br/>
<a NAME="E5:53_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<a NAME="E1:53_1_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  implies <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> ) </span>}</span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="newton.html#NM1">Prime</a><b> such that </b><br/><a NAME="E3:53_1_2_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>
<a NAME="E4:53_1_2_1"/>
( <font color="Maroon">c<sub>6</sub></font> is  <a href="int_2.html#V1">prime</a> <a href="nat_1.html#NM1">Nat</a> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="nat_4.html#E3:53_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="nat_4.html#E3:53_1_2_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E5:53_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:53_1_2"/>
(  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> ) </span>}</span>  )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:53_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="nat_4.html#E4:53_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:53"/><i><font color="Green">E54</font></i>: 
not <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_2"/><i><font color="Green">E55</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<a NAME="E2:53_2"/>
2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="int_2.html#T44">INT_2:44</a>, <a class="txt" href="nat_4.html#E1:53"><i><font color="Green">E52</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:53_2"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E1:53_2"><i><font color="Green">E55</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:53"/><i><font color="Green">E55</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  is <a href="membered.html#V2">real-membered</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="membered.html#D2">MEMBERED:def 2</a><br/>

<a NAME="E1:53_3"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  implies <font color="Maroon">c<sub>3</sub></font> is <a href="xreal_0.html#V1">real</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_3_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 ;<br/>

<a NAME="E2:53_3_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:53_3_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="xreal_0.html#V1">real</a>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:53_3"/>
(  not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  or <font color="Maroon">c<sub>3</sub></font> is <a href="xreal_0.html#V1">real</a> )
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="membered.html#V2">real-membered</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="nat_4.html#E2:53"><i><font color="Green">E53</font></i></a>, <a class="txt" href="nat_4.html#E3:53"><i><font color="Green">E54</font></i></a>, <a class="txt" href="nat_4.html#E4:53"><i><font color="Green">E55</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E6:53"/>
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="newton.html#NM1">Prime</a><b> such that </b><br/><a NAME="E8:53"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="newton.html#NM1">Prime</a> <b>by </b><i><a class="txt" href="nat_4.html#E8:53"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E10:53"/><i><font color="Green">E57</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4"><b>proof </b></a><div class="add">

<a NAME="E1:53_4"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span>  iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:53_4_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="newton.html#NM1">Prime</a><b> such that </b><br/><a NAME="E3:53_4_1_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E4:53_4_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E3:53_4_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="nat_4.html#E8:53"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E5:53_4_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="nat_4.html#E3:53_4_1_1"><i><font color="Green">E58</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:53_4_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="newton.html#NM1">Prime</a><b> such that </b><br/><a NAME="E4:53_4_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>
<a NAME="E5:53_4_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E2:53_4_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="nat_4.html#E4:53_4_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:53_4_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="nat_4.html#E4:53_4_1"><i><font color="Green">E59</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:53_4"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:53"/><i><font color="Green">E58</font></i>: 
4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_5_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="nat_4.html#E8:53"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_5_1_1"><b>suppose </b></a><a NAME="E1:53_5_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_5_1_1"/>
4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_5_1_2"><b>suppose </b></a><a NAME="E1:53_5_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:53_5_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T16">XREAL_1:16</a></i>;<br/><b>hence </b><a NAME="E3:53_5_1_2"/>
4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T44">POWER:44</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E12:53"/>
 <a href="wsierp_1.html#K9">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_6"><b>proof </b></a><div class="add">

<a NAME="E1:53_6"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E14"><i><font color="Green">Lemma14</font></i></a></i>;<br/>
<a NAME="E2:53_6"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="binarith.html#K5">-'</a> 1;<br/>
<a NAME="E3:53_6"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E2:53_6"><i><font color="Green">E59</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="wsierp_1.html#K9">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span> </span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = 4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = 4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>;<br/>
<a NAME="E4:53_6"/>
 <a href="wsierp_1.html#K9">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E3:53_6"><i><font color="Green">E60</font></i></a>, <a class="ref" href="nat_4.html#T45" target="_self">Th45</a></i>;<br/>
<b>hence </b><a NAME="E5:53_6"/>
 <a href="wsierp_1.html#K9">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E10:53"><i><font color="Green">E57</font></i></a>, <a class="txt" href="nat_4.html#E11:53"><i><font color="Green">E58</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="E13:53"/>
 <a href="wsierp_1.html#K9">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 4 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 ;<br/>


</div>
