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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:22"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:22"/><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K9">choose</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="irrat_1.html#NK1" target="_self">.^.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="irrat_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E3:22"/>
<font color="Maroon">c<sub>1</sub></font> <a href="irrat_1.html#NK1" target="_self">.^.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="irrat_1.html#E1:22"><i><font color="Green">E20</font></i></a>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/>
<b>hence </b><a NAME="E4:22"/><i><font color="Green">E22</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="irrat_1.html#E2:22"><i><font color="Green">E21</font></i></a>, <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <span class="p1">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="newton.html#K12">!</a> </span>)</span>;<br/>
<a NAME="E5:22"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="irrat_1.html#T11" target="_self">Th11</a>, <a class="ref" href="newton.html#T18">NEWTON:18</a></i>;<br/>
<div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/><b>thus </b><a NAME="E2:22_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:22_1_1_1_1"/><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="irrat_1.html#K1" target="_self">aseq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="irrat_1.html#E1:22"><i><font color="Green">E20</font></i></a>, <a class="ref" href="irrat_1.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E3:22_1_1_1_1"/>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="irrat_1.html#E1:22_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E4:22_1_1_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span>
 <b>by </b><i><a class="ref" href="irrat_1.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E5:22_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="irrat_1.html#K1" target="_self">aseq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="irrat_1.html#E1:22_1_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="irrat_1.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E6:22_1_1_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="irrat_1.html#K1" target="_self">aseq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="irrat_1.html#K1" target="_self">aseq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="irrat_1.html#E4:22_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E7:22_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="irrat_1.html#K1" target="_self">aseq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="irrat_1.html#E1:22_1_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="irrat_1.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E8:22_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="irrat_1.html#K1" target="_self">aseq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span>
 <b>by </b><i><a class="ref" href="real_2.html#T147">REAL_2:147</a></i>;<br/><b>hence </b><a NAME="E9:22_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="irrat_1.html#E2:22_1_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="irrat_1.html#E6:22_1_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:22_1_1_1_2"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><span class="p1">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K9">choose</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="irrat_1.html#NK1" target="_self">.^.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="irrat_1.html#D2" target="_self">Def2</a></i>
<br/>.= 
0 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="irrat_1.html#NK1" target="_self">.^.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="irrat_1.html#E2:22_1_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="newton.html#D3">NEWTON:def 3</a></i>
<br/>.= 
0

;<br/><b>hence </b><a NAME="E4:22_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E7:22"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <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#S1">NAT_1:sch 1</a>(<a class="txt" href="irrat_1.html#E5:22"><i><font color="Green">E23</font></i></a>, <a class="txt" href="irrat_1.html#E6:22"><i><font color="Green">E24</font></i></a>);<br/></i>
<b>hence </b><a NAME="E8:22"/><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K12">!</a> </span>)</span>
 ;<br/>

<b>hence </b><a NAME="E9:22"/>
<span class="p1">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <a href="irrat_1.html#K5" target="_self">eseq</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="irrat_1.html#D5" target="_self">Def5</a></i>;<br/>

<b>hence </b><a NAME="E10:22"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K3" target="_self">cseq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="irrat_1.html#K3" target="_self">cseq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K12">!</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="irrat_1.html#K3" target="_self">cseq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <a href="irrat_1.html#K5" target="_self">eseq</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="irrat_1.html#E4:22"><i><font color="Green">E22</font></i></a>, <a class="txt" href="irrat_1.html#E8:22"><i><font color="Green">E25</font></i></a>, <a class="ref" href="irrat_1.html#T3" target="_self">Th3</a></i>;<br/>


</div>
