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<div class="add">

<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">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/><b>let </b><font color="Maroon">a</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<a NAME="E1:65"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span>;<br/>
<a NAME="E2:65"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>thus </b><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> 0</span>)</span> = 
<font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> 1
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>
<br/>.= 
0 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:65"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">x</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:65_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>]
 ;<br/>

<b>thus </b><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span>

;<br/>


</div><b>end;</b></div>
<a NAME="E4:65"/>
for <font color="Olive">x</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E5:65"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span>
 ;<br/>


</div>
