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<b>let </b><font color="Maroon">s</font> be   <a href="nat_1.html#NM1">Nat</a>;<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>1 <a href="newton.html#K3" target="_self">|^</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1;<br/>
<a NAME="E1:17"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:17_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font>]
 ;<br/><a NAME="E2:17_1"/><b>then </b>
1 <a href="newton.html#K3" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K2">*</a> 1
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:17_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E3:17"/>
for <font color="Olive">s</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</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="E4:17"/>
1 <a href="newton.html#K3" target="_self">|^</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>


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