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<b>let </b><font color="Maroon">b</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">a</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:60"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/>

<b>thus </b><a NAME="E2:60"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 implies  not <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:60_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:60_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">b</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E3:60_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E4:60_1"/>
not <font color="Maroon">b</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E5:60_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_3.html#E1"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="newton.html#T10">NEWTON:10</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:60"/><i><font color="Green">E50</font></i>: 
not <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E4:60"/><b>then </b><i><font color="Green">E51</font></i>: 
not <font color="Maroon">b</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>;<br/>
<a NAME="E5:60"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T53">NAT_1:53</a></i>;<br/>
<a NAME="E6:60"/>
1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T53">NAT_1:53</a></i>;<br/>
<a NAME="E7:60"/><b>then </b>
<font color="Maroon">b</font> <a href="newton.html#K2">|^</a> 0 <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/>
<b>hence </b><a NAME="E8:60"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, , <a class="txt" href="nat_3.html#E1"><i><font color="Green">Lemma33</font></i></a>, </i>;<br/>


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