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<span class="kw">given </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><a NAME="E2:127"/><span class="lab"><font color="Green" title="E84">A1</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</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">consider </span><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:127"/><span class="lab"><font color="Green" title="E85">A2</font></span>: 
<font color="Maroon" title="c4">p</font> <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> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3
 <span class="kw">and </span><br/><a NAME="E5:127"/><span class="lab"><font color="Green" title="E86">A3</font></span>: 
<font color="Maroon" title="c4">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">and </span><br/><a NAME="E6:127"/><span class="lab"><font color="Green" title="E87">A4</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</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"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number05.html#T3" onmouseover="rs('number05/T3')" onmouseout="rh()">NUMBER05:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c4">p</font> as   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:127"><span class="lab"><font color="Green" title="E86">A3</font></span></a></span>;<br/>
<a NAME="E8:127"/><span class="lab"><font color="Green" title="E88">A5</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</a></span>;<br/>
<a NAME="E9:127"/><span class="lab"><font color="Green" title="E89">A6</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:127"><span class="lab"><font color="Green" title="E85">A2</font></span></a></span>;<br/>
<a NAME="E10:127"/><span class="lab"><font color="Green" title="E90">A7</font></span>: 
 <a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_1.html#T20" onmouseover="rs('euler_1/T20')" onmouseout="rh()">EULER_1:20</a></span>;<br/>
<span class="lab"><font color="Green" title="E91">A8</font></span>: <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> = 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:127"><span class="lab"><font color="Green" title="E85">A2</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T9" onmouseover="rs('newton/T9')" onmouseout="rh()">NEWTON:9</a></span>
;<br/>
<a NAME="E12:127"/><span class="lab"><font color="Green" title="E92">A9</font></span>: 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2 <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="c3">z</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T1" onmouseover="rs('wsierp_1/T1')" onmouseout="rh()">WSIERP_1:1</a></span>;<br/>
<a NAME="E13:127"/><span class="lab"><font color="Green" title="E93">A10</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:127"><span class="lab"><font color="Green" title="E84">A1</font></span></a></span>;<br/>
<a NAME="E14:127"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A11</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:127"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="ref" href="int_2.html#T2" onmouseover="rs('int_2/T2')" onmouseout="rh()">INT_2:2</a></span>;<br/>
<a NAME="E15:127"/>
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> , <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c6">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:127"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="txt" href="#E13:127"><span class="lab"><font color="Green" title="E93">A10</font></span></a>, <a class="ref" href="int_2.html#T2" onmouseover="rs('int_2/T2')" onmouseout="rh()">INT_2:2</a></span>;<br/>
<a NAME="E16:127"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c6">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_3.html#T34" onmouseover="rs('gr_cy_3/T34')" onmouseout="rh()">GR_CY_3:34</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E95">A12</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c6">p</font> = 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c6">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:127"><span class="lab"><font color="Green" title="E92">A9</font></span></a>, <a class="ref" href="nat_d.html#T64" onmouseover="rs('nat_d/T64')" onmouseout="rh()">NAT_D:64</a></span>
<br/>.= 
<font color="Maroon" title="c6">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="number10.html#T5" onmouseover="rs('number10/T5')" onmouseout="rh()">NUMBER10:5</a></span>
;<br/>
<a NAME="E18:127"/>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c6">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="127_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">w</font> be   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pythtrip.html#D2" onmouseover="rs('pythtrip/D2')" onmouseout="rh()">PYTHTRIP:def 2</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"> (  not <a href="nat_d.html#R1" title="NAT_D:pred.1">R20</a>(<font color="Maroon" title="c7">w</font>,2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">z</font>) or  not <a href="nat_d.html#R1" title="NAT_D:pred.1">R20</a>(<font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c6">p</font>) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:127_3"/><span class="lab"><font color="Green" title="E96">A13</font></span>: 
<font color="Maroon" title="c7">w</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">z</font>
 <span class="kw">and </span><br/><a NAME="E2:127_3"/><span class="lab"><font color="Green" title="E97">A14</font></span>: 
<font color="Maroon" title="c7">w</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">p</font>
 ; <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/>

<a NAME="E3:127_3"/>
<font color="Maroon" title="c7">w</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</a></span>;<br/>
<a NAME="E4:127_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:127_3"><span class="lab"><font color="Green" title="E97">A14</font></span></a>, <a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</a></span>;<br/>
<a NAME="E5:127_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_3"><span class="lab"><font color="Green" title="E96">A13</font></span></a>, <a class="ref" href="int_2.html#T2" onmouseover="rs('int_2/T2')" onmouseout="rh()">INT_2:2</a></span>;<br/>
<a NAME="E6:127_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:127"><span class="lab"><font color="Green" title="E94">A11</font></span></a>, <a class="ref" href="int_5.html#T1" onmouseover="rs('int_5/T1')" onmouseout="rh()">INT_5:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:127_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:127"><span class="lab"><font color="Green" title="E88">A5</font></span></a>, <a class="ref" href="wsierp_1.html#T15" onmouseover="rs('wsierp_1/T15')" onmouseout="rh()">WSIERP_1:15</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E19:127"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:127"><span class="lab"><font color="Green" title="E89">A6</font></span></a>, <a class="txt" href="#E10:127"><span class="lab"><font color="Green" title="E90">A7</font></span></a>, <a class="txt" href="#E11:127"><span class="lab"><font color="Green" title="E91">A8</font></span></a>, <a class="txt" href="#E17:127"><span class="lab"><font color="Green" title="E95">A12</font></span></a>, <a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</a>, <a class="ref" href="euler_2.html#T18" onmouseover="rs('euler_2/T18')" onmouseout="rh()">EULER_2:18</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/>


</div>
