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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c2">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  implies <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E2:16"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  or <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:16"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c2">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/>

<span class="kw">assume </span><a NAME="E4:16"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
 not <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c6">k</font> = <font color="Maroon" title="c3">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E6:16"/>
<font color="Maroon" title="c4">a</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="pepin.html#T2" onmouseover="rs('pepin/T2')" onmouseout="rh()">PEPIN:2</a></span>;<br/>
<a NAME="E7:16"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" onmouseover="rs('nat_d/D5')" onmouseout="rh()">NAT_D:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E9:16"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" onmouseover="rs('nat_d/D3')" onmouseout="rh()">NAT_D:def 3</a></span>;<br/>
<a NAME="E10:16"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="pepin.html#T12" onmouseover="rs('pepin/T12')" onmouseout="rh()">PEPIN:12</a></span>;<br/>
<a NAME="E11:16"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T13" onmouseover="rs('nat_d/T13')" onmouseout="rh()">NAT_D:13</a></span>;<br/>
<a NAME="E12:16"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="newton.html#T11" onmouseover="rs('newton/T11')" onmouseout="rh()">NEWTON:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:16"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T13" onmouseover="rs('nat_d/T13')" onmouseout="rh()">NAT_D:13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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