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<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> 9 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 )</span><br/>

<span class="kw">assume </span><a NAME="E1:11"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9
 ; <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 )</span><br/>

<a NAME="E2:11"/><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> 9
 
<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="E3:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">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> 9
 
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="11_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> 9 &amp;  not 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9 &amp;  not 5 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9 &amp;  not 6 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9 &amp;  not 7 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9 &amp;  not 8 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9 )</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> 4</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="E2:11_1"/>
 not 2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9
 ; <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> 9 &amp;  not 5 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9 &amp;  not 6 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9 &amp;  not 7 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9 &amp;  not 8 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9 )</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> 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> 4 = 
1 <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/>.= 
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="E4:11_1"/>
 not 4 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9
 <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> 9 &amp;  not 6 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9 &amp;  not 7 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9 &amp;  not 8 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9 )</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> 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> 5 = 
4 <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/>.= 
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="E6:11_1"/>
 not 5 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9
 <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> 9 &amp;  not 7 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9 &amp;  not 8 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9 )</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> 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> 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="E8:11_1"/>
 not 6 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9
 <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> 9 &amp;  not 8 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9 )</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> 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> 7 = 
2 <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/>.= 
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="E10:11_1"/>
 not 7 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9
 <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> 9</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> 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> 8 = 
1 <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/>.= 
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="E12:11_1"/>
 not 8 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> 9
 <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="E5:11"/>
( <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 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E23">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/>


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