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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:59"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/>

<b>thus </b><a NAME="E2:59"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 implies  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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:59_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:59_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E3:59_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E4:59_1"/>
not <font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="nat_3.html#E1:59"><i><font color="Green">E30</font></i></a>, <a class="txt" href="nat_3.html#E1:59_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="nat_3.html#E2:59_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="nat_3.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E5:59_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_3.html#E3:59_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="newton.html#T10">NEWTON:10</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:59"/><i><font color="Green">E31</font></i>: 
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>
 ;<br/>

<a NAME="E4:59"/><b>then </b><i><font color="Green">E32</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>;<br/>
<a NAME="E5:59"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="nat_3.html#E3:59"><i><font color="Green">E31</font></i></a>, <a class="ref" href="nat_1.html#T53">NAT_1:53</a></i>;<br/>
<a NAME="E6:59"/>
1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T53">NAT_1:53</a></i>;<br/>
<a NAME="E7:59"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K2">|^</a> 0 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/>
<b>hence </b><a NAME="E8:59"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="nat_3.html#E1:59"><i><font color="Green">E30</font></i></a>, <a class="txt" href="nat_3.html#E4:59"><i><font color="Green">E32</font></i></a>, <a class="txt" href="nat_3.html#E5:59"><i><font color="Green">E33</font></i></a>, <a class="ref" href="nat_3.html#D7" target="_self">Def7</a></i>;<br/>


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