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<b>let </b><font color="Maroon" title="c1">s</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ] <b>means </b>1 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1;<br/>
<a NAME="E1:17"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="newton.html#T9" target="_self" title="NEWTON:th.9">Th9</a></i>;<br/>
<div><i><font color="Green" title="E10">A2</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" title="c2">s</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:17_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">s</font>]
 ;<br/><a NAME="E2:17_1"/><b>then </b>
1 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T11" target="_self" title="NEWTON:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E3:17_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E3:17"/>
for <font color="Olive" title="b1">s</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="newton.html#E1:17"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="newton.html#E2:17"><i><font color="Green" title="E10">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:17"/>
1 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>


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