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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> ;<br/><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><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="E1:20"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="int_2.html#R2">are_relative_prime</a>  )
 ;<br/>

<a NAME="E2:20"/><b>then </b><i><font color="Green">E31</font></i>: 
( <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">b</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E3:20"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i/>;<br/>
<a NAME="E4:20"/><b>then </b><i><font color="Green">E44</font></i>: 
( <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#K10">hcf</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <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#K10">hcf</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="newton.html#T71">NEWTON:71</a></i>;<br/>
<a NAME="E5:20"/><i><font color="Green">E45</font></i>: 
( <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#K10">hcf</a> <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> &amp; <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#K10">hcf</a> <font color="Maroon">c</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<a NAME="E6:20"/><i><font color="Green">E46</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<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#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">a</font> <a href="nat_1.html#R1">divides</a> <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#K10">hcf</a> <font color="Maroon">c</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<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#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">a</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<a NAME="E7:20"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">a</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/>
<a NAME="E8:20"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<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#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">a</font> <a href="nat_1.html#R1">divides</a> 1
 
<b>by </b><i>, <a class="txt" href="euler_1.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="newton.html#T63">NEWTON:63</a></i>;<br/>
<a NAME="E9:20"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<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#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="newton.html#T71">NEWTON:71</a></i>;<br/>
<a NAME="E10:20"/>
( <span class="p1">(<span class="default"><span class="p2">(<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#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<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#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>
<a NAME="E11:20"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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#K10">hcf</a> <font color="Maroon">c</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E3"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="nat_1.html#T27">NAT_1:27</a></i>;<br/>
<a NAME="E12:20"/><b>then </b>
<font color="Maroon">a</font>,<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#K10">hcf</a> <font color="Maroon">c</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E13:20"/><b>then </b>
<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#K10">hcf</a> <font color="Maroon">c</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E14:20"/><b>then </b>
<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#K10">hcf</a> <font color="Maroon">c</font> <a href="nat_1.html#R1">divides</a> 1
 
<b>by </b><i>, , <a class="ref" href="newton.html#T63">NEWTON:63</a></i>;<br/>
<a NAME="E15:20"/><b>then </b>
( <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#K10">hcf</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <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#K10">hcf</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>
<a NAME="E16:20"/><b>then </b>
<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#K10">hcf</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T27">NAT_1:27</a></i>;<br/>
<b>hence </b><a NAME="E17:20"/>
<font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>


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