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<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:52_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:52_1_2"/><i><font color="Green">E27</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K2">|^</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="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:52_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:52_1_2"/><i><font color="Green">E29</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:52_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_2_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="nat_3.html#E5:52_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1"><b>suppose </b></a><a NAME="E1:52_1_2_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:52_1_2_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1_2_1_1_1"/>
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:52_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:52_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E4:52_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_3.html#E1:52_1_2_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="nat_3.html#E3:52_1_2_1_1"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:52_1_2_1_1"/><b>then </b>
<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>3</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:52_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K2">|^</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="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="radix_1.html#T7">RADIX_1:7</a></i>;<br/><b>hence </b><a NAME="E7:52_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_3.html#E2:52_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="nat_3.html#E3:52_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="nat_3.html#E3:52_1_2_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_2"><b>suppose </b></a><a NAME="E1:52_1_2_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:52_1_2_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1_2_1_2_1"/>
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:52_1_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:52_1_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E4:52_1_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_3.html#E1:52_1_2_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="nat_3.html#E3:52_1_2_1_2"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:52_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</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> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:52_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="radix_1.html#T7">RADIX_1:7</a></i>;<br/><b>hence </b><a NAME="E7:52_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_3.html#E1:52_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="nat_3.html#E4:52_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="nat_3.html#E3:52_1_2_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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