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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">c</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  implies <font color="Maroon" title="c3">c</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">b</font> )</span><br/>





<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E16">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">c</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">c</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">b</font></span><br/>

<a NAME="E2:14"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></i>;<br/>
<a NAME="E3:14"/><b>then </b><i><font color="Green" title="E17">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="ref" href="euler_1.html#T6" target="_self" title="EULER_1:th.6">Th6</a></i>;<br/>
<a NAME="E4:14"/>
<font color="Maroon" title="c3">c</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></i>;<br/>
<b>hence </b><a NAME="E5:14"/>
<font color="Maroon" title="c3">c</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="#E1:14"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="newton.html#T63" title="NEWTON:th.63">NEWTON:63</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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