<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:2"/><i><font color="Green" title="E3">A1</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c1">L</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b>( <font color="Maroon" title="c1">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> $1 );<br/>
<a NAME="E2:2"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="power.html#T29" title="POWER:th.29">POWER:29</a></i>;<br/>
<a NAME="E3:2"/><i><font color="Green" title="E5">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being   <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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">k</font> be    <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/>

<b>assume </b><a NAME="E1:2_1"/><i><font color="Green" title="E6">A4</font></i>: 
( <font color="Maroon" title="c1">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">k</font> )
 ;<br/>

<i><font color="Green" title="E7">A5</font></i>: <font color="Maroon" title="c1">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="tbsp_1.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
;<br/>
<a NAME="E3:2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">L</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">L</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">k</font>
 
<b>by </b><i><a class="txt" href="tbsp_1.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="tbsp_1.html#E1:2_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="real_2.html#T147" title="REAL_2:th.147">REAL_2:147</a></i>;<br/>
<b>hence </b><a NAME="E4:2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="tbsp_1.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="tbsp_1.html#E1:2_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="tbsp_1.html#E2:2_1"><i><font color="Green" title="E7">A5</font></i></a>, <a class="ref" href="real_2.html#T122" title="REAL_2:th.122">REAL_2:122</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:2"/>
for <font color="Olive" title="b1">k</font> being   <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="tbsp_1.html#E2:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="tbsp_1.html#E3:2"><i><font color="Green" title="E5">A3</font></i></a>);</i><br/>


</div>
