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<span class="kw">let </span><font color="Maroon" title="c1">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c1">i</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E1:16"/>
not <font color="Maroon" title="c1">i</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">i9</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E3:16"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">i9</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T1" title="ABIAN:th.1">ABIAN:1</a></span>;<br/>
<a NAME="E4:16"/>
<font color="Maroon" title="c1">i</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">i9</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i9</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E7">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 4 = 
1 <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 4
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_1.html#T13" title="EULER_1:th.13">EULER_1:13</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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