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<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  non <a href="xboole_0.html#V1">zero</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<a NAME="E1:10"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<b>thus </b><a NAME="E2:10"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green">E8</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="E2:10_1"/><b>then </b><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> 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="newton.html#T10">NEWTON:10</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> or  not <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1"><b>suppose </b></a><a NAME="E1:10_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:10_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/>

<a NAME="E2:10_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/>
<a NAME="E3:10_1_1_1_1"/><b>then </b>
not <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.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="moebius1.html#E1:10"><i><font color="Green">E7</font></i></a>, <a class="ref" href="nat_3.html#D7">NAT_3:def 7</a></i>;<br/>
<b>hence </b><a NAME="E4:10_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="moebius1.html#E1:10_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="newton.html#T10">NEWTON:10</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:10_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_2"><b>suppose </b></a><a NAME="E1:10_1_1_2"/>
not <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:10_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">1 <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>
 ;<br/><b>hence </b><a NAME="E3:10_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="nat_3.html#D7">NAT_3:def 7</a>, <a class="txt" href="moebius1.html#E1:10"><i><font color="Green">E7</font></i></a>, <a class="txt" href="moebius1.html#E2:10_1"><i><font color="Green">E9</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>assume </b><a NAME="E3:10"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">zero</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<a NAME="E5:10"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E6:10"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>3</sub></font> <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_3.html#D7">NAT_3:def 7</a></i>;<br/>
<a NAME="E7:10"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.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="nat_3.html#T4">NAT_3:4</a></i>;<br/>
<b>hence </b><a NAME="E8:10"/>
<font color="Maroon">c<sub>1</sub></font> <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/>


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