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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 );<br/>
<a NAME="E1:17"/>
1 <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> 0
 
;<br/>
<b>then </b>2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
2 <a href="series_1.html#K3">to_power</a> 0
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a></i>
;<br/>
<a NAME="E3:17"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
;<br/>
<a NAME="E4:17"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">k</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:17_2"/><i><font color="Green">E41</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 ;<br/>

<a NAME="E2:17_2"/>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/>
<a NAME="E3:17_2"/><b>then </b>
( 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_2.html#T11">NAT_2:11</a></i>;<br/>
<a NAME="E4:17_2"/><b>then </b>
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="power.html#T44">POWER:44</a></i>;<br/>
<b>hence </b><a NAME="E5:17_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E5:17"/>
for <font color="Olive">n</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="binarith.html#S1">BINARITH:sch 1</a>(, );</i><br/>


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