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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:27"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:27"/><i><font color="Green">E19</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="measure6.html#T23" target="_self">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E5:27"/><i><font color="Green">E20</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="measure6.html#E3:27"><i><font color="Green">E19</font></i></a>, <a class="ref" href="measure6.html#T26" target="_self">Th26</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K3" target="_self">len</a> <font color="Olive">b<sub>1</sub></font>;<br/>
<a NAME="E6:27"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a> ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/>



<b>take </b>
 <a href="measure6.html#K3" target="_self">len</a> <font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E1:27_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>, <a href="measure6.html#K3" target="_self">len</a> <font color="Maroon">c<sub>5</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E8:27"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>(<a class="txt" href="measure6.html#E6:27"><i><font color="Green">E21</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E9:27"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="txt" href="measure6.html#E5:27"><i><font color="Green">E20</font></i></a>, <a class="txt" href="measure6.html#E8:27"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E10:27"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</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="27_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:27_2"/><i><font color="Green">E25</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:27_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure6.html#E8:27"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:27_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="measure6.html#E1:27_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="measure6.html#T27" target="_self">Th27</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E11:27"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</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="measure6.html#E9:27"><i><font color="Green">E23</font></i></a>, <a class="txt" href="measure6.html#E10:27"><i><font color="Green">E24</font></i></a>);</i><br/>

<a NAME="E12:27"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <font color="Olive">b<sub>1</sub></font>
 
;<br/>
<a NAME="E13:27"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a></i>;<br/>
<a NAME="E14:27"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:27_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_3"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="measure6.html#E3:27_3"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E5:27_3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i><a class="txt" href="measure6.html#E5:27"><i><font color="Green">E20</font></i></a>, <a class="txt" href="measure6.html#E8:27"><i><font color="Green">E22</font></i></a>, <a class="ref" href="supinf_2.html#T63">SUPINF_2:63</a></i>;<br/>
<a NAME="E6:27_3"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>3</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="27_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:27_3_1"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:27_3_1"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T63">SUPINF_2:63</a></i>;<br/>
<a NAME="E3:27_3_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure6.html#E8:27"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E4:27_3_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="measure6.html#E11:27"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E5:27_3_1"/><b>then </b><i><font color="Green">E33</font></i>: 
( <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="measure6.html#E3:27_3_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="measure6.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E6:27_3_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="measure6.html#E11:27"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E7:27_3_1"/><b>then </b>
<a href="supinf_2.html#K1">0.</a>  <a href="supinf_2.html#K2">+</a> <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="measure1.html#T4">MEASURE1:4</a></i>;<br/>
<a NAME="E8:27_3_1"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/>
<a NAME="E9:27_3_1"/><i><font color="Green">E35</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="measure6.html#E13:27"><i><font color="Green">E26</font></i></a>, <a class="ref" href="supinf_2.html#T59">SUPINF_2:59</a></i>;<br/>
<a NAME="E10:27_3_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure6.html#E5:27_3_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="measure6.html#E8:27_3_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="measure1.html#T4">MEASURE1:4</a></i>;<br/>
<a NAME="E11:27_3_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="measure6.html#E11:27"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E12:27_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure6.html#E9:27_3_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="measure6.html#E10:27_3_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="measure4.html#T1">MEASURE4:1</a></i>;<br/>
<b>hence </b><a NAME="E13:27_3_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="measure6.html#E1:27_3_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="measure6.html#E2:27_3_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:27_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>3</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="measure6.html#E5:27_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="measure6.html#E6:27_3"><i><font color="Green">E29</font></i></a>);<br/></i>
<a NAME="E8:27_3"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E9:27_3"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="measure6.html#E11:27"><i><font color="Green">E25</font></i></a>, <a class="txt" href="measure6.html#E13:27"><i><font color="Green">E26</font></i></a>, <a class="ref" href="supinf_2.html#T59">SUPINF_2:59</a></i>;<br/>
<b>hence </b><a NAME="E10:27_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure6.html#E3:27_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="measure6.html#E8:27_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="measure6.html#T22" target="_self">Th22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:27"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K17">rng</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a></i>;<br/>
<a NAME="E16:27"/><b>then </b><i><font color="Green">E27</font></i>: 
(  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a>, <a class="ref" href="supinf_2.html#D23">SUPINF_2:def 23</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_4"/>
(  <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> or  <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure6.html#E16:27"><i><font color="Green">E27</font></i></a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_1"><b>suppose </b></a><a NAME="E1:27_4_1"/>
 <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_4_1"/>
 <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="measure6.html#E3:27"><i><font color="Green">E19</font></i></a>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_2"><b>suppose </b></a><a NAME="E1:27_4_2"/>
 <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_4_2"/>
 <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="measure6.html#E3:27"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
