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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</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">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">p</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  implies for <font color="Olive" title="b1">x</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  holds not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>





<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E25">A1</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">p</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</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"> for <font color="Olive" title="b1">x</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  holds not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:33"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E3:33"/><b>then </b><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<b>given </b><font color="Maroon" title="c4">x</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><a NAME="E5:33"/><i><font color="Green" title="E27">A4</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:33"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:33"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="#E5:33"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="int_4.html#T22" target="_self" title="INT_4:th.22">Th22</a></i>;<br/>
<a NAME="E7:33"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:33"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="int_4.html#T19" target="_self" title="INT_4:th.19">Th19</a></i>;<br/>
<a NAME="E8:33"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">q</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<b>hence </b><a NAME="E9:33"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:33"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="int_4.html#T21" target="_self" title="INT_4:th.21">Th21</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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