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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="radix_1.html#K1">Radix</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
 <a href="radix_1.html#K1">Radix</a> 1 = 
2 <a href="power.html#K4">to_power</a> 1

<br/>.= 
2
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/>
<a NAME="E2:4"/><b>then </b><i><font color="Green">E5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
;<br/>
<a NAME="E3:4"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:4_2"/><i><font color="Green">E7</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<i><font color="Green">E8</font></i>:  <a href="radix_1.html#K1">Radix</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>

;<br/>
<a NAME="E3:4_2"/>
 <a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="radix_3.html#E1:4_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E4:4_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="radix_3.html#E2:4_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="real_2.html#T144">REAL_2:144</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S9">NAT_1:sch 9</a>(<a class="txt" href="radix_3.html#E2:4"><i><font color="Green">E5</font></i></a>, <a class="txt" href="radix_3.html#E3:4"><i><font color="Green">E6</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/>2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="radix_1.html#K1">Radix</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>


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