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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="newton.html#K12" target="_self">!</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:56"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E2:56"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">l</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">l</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">l</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:56_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">l</font> <a href="newton.html#K12" target="_self">!</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">l</font>
 ;<br/>

<a NAME="E2:56_1"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="newton.html#K12" target="_self">!</a> </span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12" target="_self">!</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:56_1"/>
( ( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12" target="_self">!</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">l</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12" target="_self">!</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_1_1"/>
( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">l</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1"><b>case </b></a><a NAME="E1:56_1_1_1_1"/>
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:56_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12" target="_self">!</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_2"><b>case </b></a><a NAME="E1:56_1_1_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">l</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:56_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12" target="_self">!</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">l</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E3:56_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12" target="_self">!</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T3" target="_self">Th3</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:56_1"/>
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12" target="_self">!</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:56"/>
for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">l</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


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