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<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:7"/><i><font color="Green">E33</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> is <a href="int_2.html#V1">prime</a> )
 ;<br/>

<b>assume </b><a NAME="E2:7"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E3:7"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K5">div</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T47">NAT_1:47</a>, </i>;<br/>
<a NAME="E4:7"/><b>then </b>
<font color="Maroon">c</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E5:7"/><b>then </b>
( <font color="Maroon">c</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> or <font color="Maroon">c</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="newton.html#T98">NEWTON:98</a></i>;<br/>
<b>hence </b><a NAME="E6:7"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>


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