<?xml version="1.0"?>
<div class="add">

<a NAME="E1:13"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 113 &amp; <font color="Olive" title="b1">n</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  holds <br/> not <font color="Olive" title="b1">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 113
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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"> ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 113 &amp; <font color="Maroon" title="c1">n</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  implies  not <font color="Maroon" title="c1">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 113 )</span><br/>

<a NAME="E1:13_1"/>
113 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 56</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<a NAME="E2:13_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A1</font></span>: 
 not 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 113
 
;<br/>
<a NAME="E3:13_1"/>
113 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 37</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
;<br/>
<a NAME="E4:13_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A2</font></span>: 
 not 3 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 113
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_4.html#T9" onmouseover="rs('nat_4/T9')" onmouseout="rh()">NAT_4:9</a></span>;<br/>
<a NAME="E5:13_1"/>
113 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 22</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3
 
;<br/>
<a NAME="E6:13_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 not 5 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 113
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_4.html#T9" onmouseover="rs('nat_4/T9')" onmouseout="rh()">NAT_4:9</a></span>;<br/>
<a NAME="E7:13_1"/>
113 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">7 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 16</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<a NAME="E8:13_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A4</font></span>: 
 not 7 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 113
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_4.html#T9" onmouseover="rs('nat_4/T9')" onmouseout="rh()">NAT_4:9</a></span>;<br/>
<a NAME="E9:13_1"/>
113 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">11 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 10</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3
 
;<br/>
<a NAME="E10:13_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A5</font></span>: 
 not 11 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 113
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_4.html#T9" onmouseover="rs('nat_4/T9')" onmouseout="rh()">NAT_4:9</a></span>;<br/>
<a NAME="E11:13_1"/>
113 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">13 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 8</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 9
 
;<br/>
<a NAME="E12:13_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A6</font></span>: 
 not 13 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 113
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_4.html#T9" onmouseover="rs('nat_4/T9')" onmouseout="rh()">NAT_4:9</a></span>;<br/>
<a NAME="E13:13_1"/>
113 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">17 <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> 11
 
;<br/>
<a NAME="E14:13_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A7</font></span>: 
 not 17 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 113
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_4.html#T9" onmouseover="rs('nat_4/T9')" onmouseout="rh()">NAT_4:9</a></span>;<br/>
<a NAME="E15:13_1"/>
113 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">19 <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> 18
 
;<br/>
<a NAME="E16:13_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A8</font></span>: 
 not 19 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 113
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_4.html#T9" onmouseover="rs('nat_4/T9')" onmouseout="rh()">NAT_4:9</a></span>;<br/>
<a NAME="E17:13_1"/>
113 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">23 <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> 21
 
;<br/>
<a NAME="E18:13_1"/><span class="kw">then </span>
 not 23 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 113
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_4.html#T9" onmouseover="rs('nat_4/T9')" onmouseout="rh()">NAT_4:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:13_1"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 113 &amp; <font color="Maroon" title="c1">n</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  implies  not <font color="Maroon" title="c1">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 113 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_1"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E4:13_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E6:13_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E8:13_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E10:13_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E12:13_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E14:13_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E16:13_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="nat_4.html#T62" onmouseover="rs('nat_4/T62')" onmouseout="rh()">NAT_4:62</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="E2:13"/>
113 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_4.html#T14" onmouseover="rs('nat_4/T14')" onmouseout="rh()">NAT_4:14</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>
