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<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> ] means <font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K2" target="_self">|^</a> <font color="Maroon">a<sub>1</sub></font> is <a href="ordinal1.html#V7">natural</a>;<br/>
<a NAME="E1:13_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="newton.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E2:13_1_1"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:13_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>3</sub></font> as  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<a NAME="E3:13_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="newton.html#T11" target="_self">Th11</a></i>;<br/>
<b>hence </b><a NAME="E4:13_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:13_1_1"/>
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(<a class="txt" href="newton.html#E1:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="newton.html#E2:13_1_1"><i><font color="Green">E9</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:13_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font> is <a href="ordinal1.html#V7">natural</a>
 ;<br/>


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