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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="newton.html#NM1">Prime</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:20"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> st <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/> not <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a<sub>1</sub></font> <a href="newton.html#K12">!</a> ;<br/>
<a NAME="E2:20"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="newton.html#NM1">Prime</a>;<br/>

<b>assume </b><a NAME="E1:20_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:20_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> 0 <a href="newton.html#K12">!</a> 
 ;<br/>

<a NAME="E3:20_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T54">NAT_1:54</a>, <a class="ref" href="newton.html#T18">NEWTON:18</a></i>;<br/>
<b>hence </b><a NAME="E4:20_1"/>
contradiction
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:20"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:20_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>

<a NAME="E2:20_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> st <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/> not <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="newton.html#NM1">Prime</a>;<br/>

<b>assume </b><a NAME="E1:20_2_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:20_2_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> 
 ;<br/>

<a NAME="E3:20_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E4:20_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:20_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E5:20_2_1"/><b>then </b><i><font color="Green">E27</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K12">!</a> 
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:20_2"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E6:20_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E2:20_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="newton.html#T21">NEWTON:21</a></i>;<br/>
<a NAME="E7:20_2_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E5:20_2_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="newton.html#T98">NEWTON:98</a></i>;<br/>
<b>hence </b><a NAME="E8:20_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E1:20_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:20"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="nat_4.html#E2:20"><i><font color="Green">E22</font></i></a>, <a class="txt" href="nat_4.html#E3:20"><i><font color="Green">E23</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:20"/>
not <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K12">!</a> 
 <b>by </b><i><a class="txt" href="nat_4.html#E1:20"><i><font color="Green">E21</font></i></a></i>;<br/>


</div>
