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<div class="add">

<b>let </b><font color="Maroon" title="c1">s</font>, <font color="Maroon" title="c2">r</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="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">r</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  $1 <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Olive" title="b1">r</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>
<a NAME="E1:38"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">r</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<a NAME="E1:38_1"/>
( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
;<br/>
<b>hence </b><a NAME="E2:38_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c3">r</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="newton.html#D3" target="_self" title="NEWTON:def.3">Def3</a>, <a class="ref" href="newton.html#T27" target="_self" title="NEWTON:th.27">Th27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:38"/><i><font color="Green" title="E24">A2</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">s</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:38_2"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">s</font>]
 ;<br/>

<a NAME="E2:38_2"/><i><font color="Green" title="E26">A4</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Olive" title="b1">r</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">r</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:38_2_1"/><i><font color="Green" title="E27">A5</font></i>: 
<font color="Maroon" title="c4">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E2:38_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c4">r</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c5">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E2:38_2_1_1"/><i><font color="Green" title="E28">A6</font></i>: 
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="newton.html#E1:38_2_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">t</font> = <font color="Maroon" title="c5">t</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">m1</font> = <font color="Maroon" title="c3">s</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c6">t</font>, <font color="Maroon" title="c8">m2</font> = <font color="Maroon" title="c3">s</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="newton.html#E1:38_2"><i><font color="Green" title="E25">A3</font></i></a></i>;<br/>
<a NAME="E5:38_2_1_1"/>
<font color="Maroon" title="c7">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E6:38_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c4">r</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="newton.html#E2:38_2_1_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="newton.html#T32" target="_self" title="NEWTON:th.32">Th32</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:38_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c4">r</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon" title="c4">r</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<a NAME="E3:38_2"/>
( <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
;<br/>
<b>hence </b><a NAME="E4:38_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c4">r</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="newton.html#E2:38_2"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="newton.html#T29" target="_self" title="NEWTON:th.29">Th29</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:38"/>
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#E1:38"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="newton.html#E2:38"><i><font color="Green" title="E24">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:38"/>
<font color="Maroon" title="c1">s</font> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c2">r</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>


</div>
