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<span class="kw">let </span><font color="Maroon" title="c2">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">  not 8 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:117_1"/>
( <font color="Maroon" title="c2">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  or <font color="Maroon" title="c2">n</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1"><span class="kw">suppose </span></a><a NAME="E1:117_1_1"/>
<font color="Maroon" title="c2">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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 8 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:117_1_1"/><span class="lab"><font color="Green" title="E93">A1</font></span>: 
<font color="Maroon" title="c2">n</font> <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">k</font></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="ref" href="abian.html#T9" onmouseover="rs('abian/T9')" onmouseout="rh()">ABIAN:9</a></span>;<br/><a NAME="E4:117_1_1"/><span class="lab"><font color="Green" title="E94">A2</font></span>: 
3 <a href="newton.html#K11" title="NEWTON:func.11">|^</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">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">3 <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="c3">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T6" onmouseover="rs('newton/T6')" onmouseout="rh()">NEWTON:6</a></span>;<br/><a NAME="E5:117_1_1"/>
3 <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="c3">k</font></span>)</span>,1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 8
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T65" target="_self" onmouseover="rs('number08/T65')" onmouseout="rh()">Th65</a></span>;<br/><a NAME="E6:117_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">3 <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="c3">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3,1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 3 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 8
 <span class="kw">by</span> <span class="lab"><a class="txt" href="number08.html#E14"><span class="lab"><font color="Green" title="E14">Lm14</font></span></a>, <a class="ref" href="int_1.html#T18" onmouseover="rs('int_1/T18')" onmouseout="rh()">INT_1:18</a></span>;<br/><a NAME="E7:117_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">3 <a href="newton.html#K11" title="NEWTON:func.11">|^</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="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,4 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 8
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:117_1_1"><span class="lab"><font color="Green" title="E94">A2</font></span></a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 8 = 
4 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 8
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:117_1_1"><span class="lab"><font color="Green" title="E93">A1</font></span></a>, <a class="ref" href="nat_d.html#T64" onmouseover="rs('nat_d/T64')" onmouseout="rh()">NAT_D:64</a></span>
<br/>.= 
4
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</a></span>
;<br/><span class="kw">hence </span><a NAME="E9:117_1_1"/>
 not 8 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></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="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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_2"><span class="kw">suppose </span></a><a NAME="E1:117_1_2"/>
<font color="Maroon" title="c2">n</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</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 8 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:117_1_2"/><span class="lab"><font color="Green" title="E93">A3</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font>
 ;<br/><a NAME="E4:117_1_2"/>
3 <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="c3">k</font></span>)</span>,1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 8
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T65" target="_self" onmouseover="rs('number08/T65')" onmouseout="rh()">Th65</a></span>;<br/><a NAME="E5:117_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">3 <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="c3">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 8
 ;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 8 = 
2 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 8
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:117_1_2"><span class="lab"><font color="Green" title="E93">A3</font></span></a>, <a class="ref" href="nat_d.html#T64" onmouseover="rs('nat_d/T64')" onmouseout="rh()">NAT_D:64</a></span>
<br/>.= 
2
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:117_1_2"/>
 not 8 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></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="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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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