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<b>let </b><font color="Maroon">p1</font> be   <a href="newton.html#NM1">Prime</a>, <font color="Maroon">p2</font> be   <a href="newton.html#NM1">Prime</a>;<br/><b>let </b><font color="Maroon">m</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:23"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">p1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E2:23"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">p1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>assume </b><a NAME="E3:23"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font>
 ;<br/>

<a NAME="E4:23"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">p1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E5:23"/><b>then </b>
<font color="Maroon">p1</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i>, <a class="ref" href="power.html#T40">POWER:40</a></i>;<br/>
<a NAME="E6:23"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">p1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="power.html#T46">POWER:46</a></i>;<br/>
<a NAME="E7:23"/>
not <font color="Maroon">p1</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p2</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="nat_3.html#T6">NAT_3:6</a></i>;<br/>
<a NAME="E8:23"/><b>then </b>
<font color="Maroon">p1</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , <a class="ref" href="nat_3.html#T27">NAT_3:27</a></i>;<br/>
<a NAME="E9:23"/><b>then </b>
<font color="Maroon">p1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="nat_3.html#T25">NAT_3:25</a></i>;<br/>
<b>hence </b><a NAME="E10:23"/>
contradiction
 <b>by </b><i>, <a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/>


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