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<span class="kw">let </span><font color="Maroon" title="c17">k</font>, <font color="Maroon" title="c18">n</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"> ( <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">28 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 implies 29 <a href="nat_d.html#R1" title="NAT_D: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="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2</span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:103"/><span class="lab"><font color="Green" title="E82">A1</font></span>: 
<font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">28 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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"> 29 <a href="nat_d.html#R1" title="NAT_D: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="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2</span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c19">x</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2</span>)</span>;<br/>
<a NAME="E2:103"/><span class="lab"><font color="Green" title="E83">A2</font></span>: 
 <a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> 29 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 29 <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>, <a class="ref" href="xprimes1.html#T29" onmouseover="rs('xprimes1/T29')" onmouseout="rh()">XPRIMES1:29</a></span>;<br/>
<a NAME="E3:103"/><span class="lab"><font color="Green" title="E84">A3</font></span>: 
<span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 28</span>)</span> <a href="newton.html#K1" title="NEWTON:func.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="c17">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default">28 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">k</font></span>)</span></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="E4:103"/>
<span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> 29</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 29 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#T18" onmouseover="rs('euler_2/T18')" onmouseout="rh()">EULER_2:18</a>, <a class="ref" href="xprimes1.html#T2" onmouseover="rs('xprimes1/T2')" onmouseout="rh()">XPRIMES1:2</a>, <a class="ref" href="xprimes1.html#T29" onmouseover="rs('xprimes1/T29')" onmouseout="rh()">XPRIMES1:29</a>, <a class="ref" href="int_2.html#T30" onmouseover="rs('int_2/T30')" onmouseout="rh()">INT_2:30</a></span>;<br/>
<a NAME="E5:103"/><span class="kw">then </span>
2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 28,1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 29
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:103"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="ref" href="pepin.html#T39" onmouseover="rs('pepin/T39')" onmouseout="rh()">PEPIN:39</a></span>;<br/>
<a NAME="E6:103"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 28</span>)</span> <a href="newton.html#K1" title="NEWTON:func.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="c17">k</font></span>)</span>,1 <a href="newton.html#K1" title="NEWTON:func.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="c17">k</font></span>)</span> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 29
 
<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/>
<a NAME="E7:103"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A4</font></span>: 
<span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default">28 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 4,1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 4 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 29
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:103"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="ref" href="int_4.html#T11" onmouseover="rs('int_4/T11')" onmouseout="rh()">INT_4:11</a></span>;<br/>
2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">28 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> = 
2 <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> <span class="p3">(<span class="default">28 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2</span>)</span>

<br/>.= 
<span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default">28 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T8" onmouseover="rs('newton/T8')" onmouseout="rh()">NEWTON:8</a></span>
;<br/>
<a NAME="E9:103"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1,4 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 29
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:103"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E7:103"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="number07.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<a NAME="E10:103"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2,5 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 29
 
<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/>
<a NAME="E11:103"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2</span>)</span>,29 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 29
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="number07.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="txt" href="number07.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 29 = 
29 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 29
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T64" onmouseover="rs('nat_d/T64')" onmouseout="rh()">NAT_D:64</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T25" onmouseover="rs('nat_d/T25')" onmouseout="rh()">NAT_D:25</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:103"/>
29 <a href="nat_d.html#R1" title="NAT_D: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="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T62" onmouseover="rs('int_1/T62')" onmouseout="rh()">INT_1:62</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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