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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">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 <font color="Maroon" title="c1">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 9 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 27 )</span><br/>

<span class="kw">assume </span><a NAME="E1:13"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 ; <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="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 9 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 27 )</span><br/>

<a NAME="E2:13"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 27
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T27" onmouseover="rs('int_2/T27')" onmouseout="rh()">INT_2:27</a></span>;<br/>
<a NAME="E3:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A2</font></span>: 
 not  not <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  ...  &amp;  not <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 27
 
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_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 2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 5 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 6 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 7 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 8 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 10 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 11 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 12 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 13 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 14 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 15 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 16 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 17 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 18 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 19 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 20 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 21 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 22 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 23 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span></a><div class="add"><span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 13</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> 2 = 
1 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
1
<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="E2:13_1"/>
 not 2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 ; <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 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 5 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 6 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 7 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 8 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 10 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 11 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 12 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 13 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 14 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 15 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 16 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 17 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 18 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 19 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 20 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 21 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 22 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 23 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 6</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 = 
3 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
3
<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="E4:13_1"/>
 not 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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"> (  not 5 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 6 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 7 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 8 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 10 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 11 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 12 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 13 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 14 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 15 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 16 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 17 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 18 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 19 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 20 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 21 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 22 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 23 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">5 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 5</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 5 = 
2 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 5
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</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="E6:13_1"/>
 not 5 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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"> (  not 6 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 7 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 8 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 10 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 11 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 12 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 13 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 14 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 15 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 16 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 17 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 18 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 19 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 20 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 21 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 22 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 23 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">6 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 4</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 6 = 
3 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 6
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
3
<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="E8:13_1"/>
 not 6 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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"> (  not 7 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 8 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 10 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 11 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 12 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 13 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 14 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 15 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 16 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 17 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 18 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 19 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 20 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 21 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 22 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 23 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">7 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 3</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 6</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 7 = 
6 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 7
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
6
<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="E10:13_1"/>
 not 7 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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"> (  not 8 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 10 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 11 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 12 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 13 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 14 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 15 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 16 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 17 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 18 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 19 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 20 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 21 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 22 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 23 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">8 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 3</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 8 = 
3 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 8
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
3
<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="E12:13_1"/>
 not 8 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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"> (  not 10 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 11 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 12 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 13 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 14 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 15 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 16 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 17 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 18 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 19 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 20 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 21 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 22 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 23 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">10 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 7</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 10 = 
7 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 10
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
7
<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="E14:13_1"/>
 not 10 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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"> (  not 11 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 12 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 13 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 14 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 15 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 16 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 17 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 18 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 19 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 20 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 21 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 22 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 23 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">11 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 5</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 11 = 
5 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 11
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
5
<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="E16:13_1"/>
 not 11 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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"> (  not 12 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 13 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 14 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 15 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 16 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 17 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 18 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 19 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 20 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 21 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 22 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 23 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">12 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 12 = 
3 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 12
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
3
<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="E18:13_1"/>
 not 12 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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"> (  not 13 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 14 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 15 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 16 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 17 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 18 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 19 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 20 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 21 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 22 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 23 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">13 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</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> 13 = 
1 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 13
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
1
<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="E20:13_1"/>
 not 13 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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"> (  not 14 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 15 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 16 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 17 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 18 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 19 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 20 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 21 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 22 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 23 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">14 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 13</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 14 = 
13 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 14
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
13
<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="E22:13_1"/>
 not 14 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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"> (  not 15 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 16 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 17 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 18 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 19 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 20 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 21 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 22 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 23 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">15 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 12</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 15 = 
12 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 15
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
12
<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="E24:13_1"/>
 not 15 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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"> (  not 16 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 17 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 18 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 19 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 20 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 21 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 22 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 23 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">16 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 11</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 16 = 
11 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 16
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
11
<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="E26:13_1"/>
 not 16 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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"> (  not 17 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 18 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 19 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 20 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 21 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 22 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 23 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">17 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 10</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 17 = 
10 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 17
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
10
<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="E28:13_1"/>
 not 17 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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"> (  not 18 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 19 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 20 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 21 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 22 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 23 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">18 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 9</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 18 = 
9 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 18
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
9
<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="E30:13_1"/>
 not 18 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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"> (  not 19 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 20 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 21 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 22 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 23 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">19 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 8</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 19 = 
8 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 19
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
8
<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="E32:13_1"/>
 not 19 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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"> (  not 20 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 21 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 22 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 23 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">20 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 7</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 20 = 
7 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 20
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
7
<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="E34:13_1"/>
 not 20 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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"> (  not 21 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 22 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 23 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">21 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 6</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 21 = 
6 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 21
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
6
<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="E36:13_1"/>
 not 21 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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"> (  not 22 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 23 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">22 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 5</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 22 = 
5 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 22
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
5
<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="E38:13_1"/>
 not 22 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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"> (  not 23 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">23 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 23 = 
4 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 23
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</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="E40:13_1"/>
 not 23 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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"> (  not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">24 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 24 = 
3 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 24
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
3
<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="E42:13_1"/>
 not 24 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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"> (  not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 &amp;  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27 )</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">25 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 25 = 
2 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 25
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</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="E44:13_1"/>
 not 25 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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">  not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27</span><br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">26 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</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> 26 = 
1 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 26
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
1
<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="E46:13_1"/>
 not 26 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 27
 <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>
<span class="kw">hence </span><a NAME="E5:13"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 9 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 27 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E3:13"><span class="lab"><font color="Green" title="E25">A2</font></span></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>
