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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">  for <font color="Olive" title="b1">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st 2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Olive" title="b1">i</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 holds <br/>2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Olive" title="b1">i</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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"> ( 2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">i</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 implies 2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">i</font> )</span><br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_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="E1:141_1"/>
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1_1"><span class="kw">suppose </span></a><a NAME="E1:141_1_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"> ( 2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">i</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 implies 2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">i</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:141_1_1"/>
( 2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">i</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 implies 2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T12" onmouseover="rs('int_2/T12')" onmouseout="rh()">INT_2:12</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="141_1_2"><span class="kw">suppose </span></a><a NAME="E1:141_1_2"/><span class="lab"><font color="Green" title="E89">A1</font></span>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"> ( 2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">i</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 implies 2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">i</font> )</span><br/><div class="add"><span class="kw">assume </span><a NAME="E2:141_1_2"/><span class="lab"><font color="Green" title="E90">A2</font></span>: 
2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</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"> (  not 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">i</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 or 2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">i</font> )</span><br/><span class="kw">assume </span><a NAME="E3:141_1_2"/>
2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">i</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3
 ; <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"> 2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">i</font></span><br/><a NAME="E4:141_1_2"/><span class="kw">then </span>
2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#T4" target="_self" onmouseover="rs('number14/T4')" onmouseout="rh()">Th4</a></span>;<br/><a NAME="E5:141_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A3</font></span>: 
2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="taylor_2.html#T1" onmouseover="rs('taylor_2/T1')" onmouseout="rh()">TAYLOR_2:1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c4">a</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>, <font color="Maroon" title="c5">b</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E7:141_1_2"/><span class="lab"><font color="Green" title="E92">A4</font></span>: 
<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:141_1_2"><span class="lab"><font color="Green" title="E89">A1</font></span></a>, <a class="ref" href="nat_6.html#T3" onmouseover="rs('nat_6/T3')" onmouseout="rh()">NAT_6:3</a></span>;<br/><a NAME="E8:141_1_2"/><span class="lab"><font color="Green" title="E93">A5</font></span>: 
<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</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">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:141_1_2"><span class="lab"><font color="Green" title="E92">A4</font></span></a>, <a class="ref" href="newton.html#T7" onmouseover="rs('newton/T7')" onmouseout="rh()">NEWTON:7</a></span>;<br/><a NAME="E9:141_1_2"/><span class="lab"><font color="Green" title="E94">A6</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3,2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#T3" onmouseover="rs('nat_5/T3')" onmouseout="rh()">NAT_5:3</a></span>;<br/><a NAME="E10:141_1_2"/>
<span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 <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">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">a</font></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="E11:141_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:141_1_2"><span class="lab"><font color="Green" title="E91">A3</font></span></a>, <a class="txt" href="#E8:141_1_2"><span class="lab"><font color="Green" title="E93">A5</font></span></a>, <a class="txt" href="#E9:141_1_2"><span class="lab"><font color="Green" title="E94">A6</font></span></a>, <a class="ref" href="int_2.html#T25" onmouseover="rs('int_2/T25')" onmouseout="rh()">INT_2:25</a>, <a class="ref" href="pepin.html#T31" onmouseover="rs('pepin/T31')" onmouseout="rh()">PEPIN:31</a></span>;<br/><a NAME="E12:141_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A7</font></span>: 
2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:141_1_2"><span class="lab"><font color="Green" title="E90">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="141_1_2_1"><span class="kw">now </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">  not <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:141_1_2_1"/>
<font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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"> contradiction</span><br/><a NAME="E2:141_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:141_1_2_1"/><span class="kw">then </span>
3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><span class="kw">hence </span><a NAME="E4:141_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:141_1_2"><span class="lab"><font color="Green" title="E95">A7</font></span></a>, <a class="txt" href="number14.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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><a NAME="E14:141_1_2"/><span class="kw">then </span>
2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:141_1_2"><span class="lab"><font color="Green" title="E92">A4</font></span></a>, <a class="ref" href="int_2.html#T2" onmouseover="rs('int_2/T2')" onmouseout="rh()">INT_2:2</a>, <a class="ref" href="pepin.html#T31" onmouseover="rs('pepin/T31')" onmouseout="rh()">PEPIN:31</a></span>;<br/><span class="kw">hence </span><a NAME="E15:141_1_2"/>
2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#T4" target="_self" onmouseover="rs('number14/T4')" onmouseout="rh()">Th4</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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