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<span class="kw">let </span><font color="Maroon" title="c1">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> </span><br/>

<font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c1">k</font>

<br/>.= 
1 <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c1">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_1.html#T8" title="EULER_1:th.8">EULER_1:8</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T51" title="NEWTON:th.51">NEWTON:51</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:5"/>
<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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