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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">x</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>  st <font color="Maroon" title="c1">a</font>,<font color="Olive" title="b1">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  &amp; <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</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">a</font>,<font color="Maroon" title="c3">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  &amp; <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c3">p</font> implies <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:41"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <span class="kw">and </span><br/><a NAME="E2:41"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c3">p</font>
 ; <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="c2">x</font>,<font color="Maroon" title="c3">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> </span><br/>

<a NAME="E3:41"/>
<font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">p</font> <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="txt" href="#E1:41"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="gr_cy_3.html#T31" target="_self" title="GR_CY_3:th.31">Th31</a></span>;<br/>
<a NAME="E4:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">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="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:41"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
 not <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:41"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T21" title="INT_2:th.21">INT_2:21</a></span>;<br/>
<a NAME="E7:41"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E8:41"/>
<font color="Maroon" title="c2">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T2" title="INT_2:th.2">INT_2:2</a>, <a class="ref" href="int_2.html#T21" title="INT_2:th.21">INT_2:21</a></span>;<br/>
<a NAME="E9:41"/><span class="kw">then </span>
( <font color="Maroon" title="c2">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c2">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E6:41"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="int_2.html#T13" title="INT_2:th.13">INT_2:13</a>, <a class="ref" href="int_2.html#T22" title="INT_2:th.22">INT_2:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:41"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E7:41"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <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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