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

<b>let </b><font color="Maroon">G</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> ;<br/>

<b>assume </b><a NAME="E1:12"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">G</font> is <a href="supinf_2.html#V6">nonnegative</a>
 ;<br/>

<b>let </b><font color="Maroon">S</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">H</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">S</font>, <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E2:12"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">H</font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">F</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>   st <font color="Olive">F</font> is <a href="supinf_2.html#V6">nonnegative</a> holds <br/><span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Olive">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">a<sub>1</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="Olive">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">m</font>;<br/>
<a NAME="E3:12"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_1"/>
( 0 <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> or  not 0 <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1"><b>suppose </b></a><a NAME="E1:12_1_1_1"/><i><font color="Green">E30</font></i>: 
0 <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> 0 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>take </b>
<font color="Maroon">m</font>
;<br/><b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>assume </b><a NAME="E3:12_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">F</font> is <a href="supinf_2.html#V6">nonnegative</a>
 ;<br/><a NAME="E4:12_1_1_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0
 <b>by </b><i><a class="ref" href="supinf_2.html#T63">SUPINF_2:63</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="supinf_2.html#K13">.</a> 0
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/><b>hence </b><a NAME="E6:12_1_1_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0 <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">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2"><b>suppose </b></a><a NAME="E1:12_1_1_2"/><i><font color="Green">E33</font></i>: 
not 0 <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">m</font> = 0;<br/><b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>assume </b><a NAME="E2:12_1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">F</font> is <a href="supinf_2.html#V6">nonnegative</a>
 ;<br/><a NAME="E3:12_1_1_2"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0
 <b>by </b><i><a class="ref" href="supinf_2.html#T63">SUPINF_2:63</a></i>;<br/><a NAME="E4:12_1_1_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E5:12_1_1_2"/><i><font color="Green">E42</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F</font> <a href="measure1.html#K1">.</a> 0
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a></i>;<br/><a NAME="E6:12_1_1_2"/>
<font color="Maroon">F</font> <a href="measure1.html#K1">.</a> 0 <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">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E7:12_1_1_2"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0 <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">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:12"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">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="12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>given </b><font color="Maroon">m0</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:12_2"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">F</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>   st <font color="Olive">F</font> is <a href="supinf_2.html#V6">nonnegative</a> holds <br/><span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Olive">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">k</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="Olive">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m0</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_2_1"/>
( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> or  not <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1"><b>suppose </b></a><a NAME="E1:12_2_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>set </b><font color="Maroon">m</font> = <font color="Maroon">m0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font>;<br/><a NAME="E3:12_2_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">m0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E4:12_2_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>take </b>
<font color="Maroon">m0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font>
;<br/><b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>assume </b><a NAME="E5:12_2_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">F</font> is <a href="supinf_2.html#V6">nonnegative</a>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="measure1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) );<br/><a NAME="E6:12_2_1_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_2_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">F</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
;<br/><b>thus </b><a NAME="E2:12_2_1_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">F</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>]
 <b>by </b><i><a class="ref" href="measure7.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:12_2_1_1_1_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>take </b>
 <a href="supinf_2.html#K1">0.</a> 
;<br/><b>thus </b><a NAME="E2:12_2_1_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>, <a href="supinf_2.html#K1">0.</a> ]
 <b>by </b><i><a class="ref" href="measure7.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">G0</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="E8:12_2_1_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Maroon">G0</font> <a href="measure1.html#K1">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i><a NAME="E9:12_2_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G0</font> <a href="measure1.html#K1">.</a> <font color="Olive">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_2_1_1_2_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or  not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:12_2_1_1_2_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:12_2_1_1_2_1_1"/><b>then </b>
<font color="Maroon">G0</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="measure7.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_2_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G0</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:12_2_1_1_2_1_2"/>
not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_2_1_1_2_1_2"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G0</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="measure7.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E10:12_2_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">G0</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/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="measure1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) );<br/><a NAME="E11:12_2_1_1"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_2_1_1_3_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:12_2_1_1_3_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">F</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
;<br/><b>thus </b><a NAME="E2:12_2_1_1_3_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">F</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>]
 <b>by </b><i><a class="txt" href="measure7.html#E1:12"><i><font color="Green">E25</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:12_2_1_1_3_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>take </b>
 <a href="supinf_2.html#K1">0.</a> 
;<br/><b>thus </b><a NAME="E2:12_2_1_1_3_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>, <a href="supinf_2.html#K1">0.</a> ]
 <b>by </b><i><a class="txt" href="measure7.html#E1:12"><i><font color="Green">E25</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">G1</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="E13:12_2_1_1"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>,<font color="Maroon">G1</font> <a href="measure1.html#K1">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i><a NAME="E14:12_2_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G1</font> <a href="measure1.html#K1">.</a> <font color="Olive">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_2_1_1_4_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:12_2_1_1_4_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:12_2_1_1_4_1_1"/><b>then </b>
<font color="Maroon">G1</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="measure7.html#E1:12"><i><font color="Green">E25</font></i></a></i>;<br/><b>hence </b><a NAME="E3:12_2_1_1_4_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G1</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:12_2_1_1_4_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_2_1_1_4_1_2"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G1</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="measure7.html#E1:12"><i><font color="Green">E25</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E15:12_2_1_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">G1</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/><b>reconsider </b><font color="Maroon">GG0</font> =  <a href="measure7.html#K2" target="_self">On</a> <span class="p1">(<span class="default"><font color="Maroon">G0</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a>  ;<br/><a NAME="E17:12_2_1_1"/>
<font color="Maroon">G0</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i>, <a class="ref" href="measure1.html#T54">MEASURE1:54</a></i>;<br/><a NAME="E18:12_2_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">GG0</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><i><font color="Green">E81</font></i>: <font color="Maroon">GG0</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">G0</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">G0</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="measure7.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="measure7.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="measure7.html#T1" target="_self">Th1</a></i>
;<br/><a NAME="E21:12_2_1_1"/><i><font color="Green">E82</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 holds <br/><font color="Maroon">GG0</font> <a href="measure1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:12_2_1_1_6"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_2_1_1_6_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> or  not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_6_1_1"><b>suppose </b></a><a NAME="E1:12_2_1_1_6_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green">E85</font></i>: <font color="Maroon">GG0</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">G0</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">G0</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><a NAME="E4:12_2_1_1_6_1_1"/>
not <font color="Maroon">H</font> <a href="measure7.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , , <a class="ref" href="funct_2.html#T25">FUNCT_2:25</a></i>;<br/><b>hence </b><a NAME="E5:12_2_1_1_6_1_1"/>
<font color="Maroon">GG0</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="measure7.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_6_1_2"><b>suppose </b></a><a NAME="E1:12_2_1_1_6_1_2"/>
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_2_1_1_6_1_2"/>
<font color="Maroon">GG0</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>set </b><font color="Maroon">GG1</font> =  <a href="measure7.html#K2" target="_self">On</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span>;<br/><a NAME="E22:12_2_1_1"/>
<font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i>, <a class="ref" href="measure1.html#T54">MEASURE1:54</a></i>;<br/><a NAME="E23:12_2_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
 <a href="measure7.html#K2" target="_self">On</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i/>;<br/><i><font color="Green">E88</font></i>: <span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">G1</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="measure7.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
 <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i><a class="txt" href="measure7.html#E1:12"><i><font color="Green">E25</font></i></a></i>
;<br/><a NAME="E25:12_2_1_1"/><i><font color="Green">E89</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:12_2_1_1_8"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 ;<br/>

<b>then </b><i><font color="Green">E91</font></i>: <span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">G1</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E4:12_2_1_1_8"/>
not <font color="Maroon">H</font> <a href="measure7.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="funct_2.html#T25">FUNCT_2:25</a></i>;<br/>
<b>hence </b><a NAME="E5:12_2_1_1_8"/>
<span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure7.html#E1:12"><i><font color="Green">E25</font></i></a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E26:12_2_1_1"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">GG0</font> <a href="measure1.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_2_1_1_9_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_9_1_1"><b>suppose </b></a><a NAME="E1:12_2_1_1_9_1_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">GG0</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="measure7.html#E3:12"><i><font color="Green">E28</font></i></a>, , , <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/><b>hence </b><a NAME="E3:12_2_1_1_9_1_1"/>
<span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">GG0</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_9_1_2"><b>suppose </b></a><a NAME="E1:12_2_1_1_9_1_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_9_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_9_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_2_1_1_9_1_2_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> or  not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_9_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:12_2_1_1_9_1_2_1_1_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><div class="add"><a NAME="E2:12_2_1_1_9_1_2_1_1_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">GG0</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:12_2_1_1_9_1_2_1_1_1"/><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">GG0</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="measure7.html#T2" target="_self">Th2</a>, , <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/><b>hence </b><a NAME="E5:12_2_1_1_9_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">GG0</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_9_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:12_2_1_1_9_1_2_1_1_2"/><i><font color="Green">E98</font></i>: 
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><div class="add"><a NAME="E2:12_2_1_1_9_1_2_1_1_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">GG0</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:12_2_1_1_9_1_2_1_1_2"/><i><font color="Green">E100</font></i>: 
<span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> = 
 <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">GG0</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="measure7.html#T2" target="_self">Th2</a>, , <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/><b>hence </b><a NAME="E5:12_2_1_1_9_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">GG0</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:12_2_1_1_9_1_2"/>
<span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">GG0</font> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E27:12_2_1_1"/><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">GG0</font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="measure7.html#E3:12"><i><font color="Green">E28</font></i></a>, , </i>;<br/><a NAME="E28:12_2_1_1"/><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">G0</font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G0</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="measure7.html#T1" target="_self">Th1</a>, </i>;<br/><b>set </b><font color="Maroon">v</font> = <font color="Maroon">H</font> <a href="measure7.html#K1" target="_self">.</a> <font color="Maroon">n</font>;<br/><a NAME="E29:12_2_1_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">G0</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="measure7.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="measure7.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="measure7.html#T1" target="_self">Th1</a></i>;<br/><i><font color="Green">E105</font></i>: <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p4">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</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/>.= 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/><a NAME="E31:12_2_1_1"/><i><font color="Green">E106</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">k</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">G1</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m0</font>
 <b>by </b><i>, </i>;<br/><a NAME="E32:12_2_1_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">G1</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m0</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">G1</font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E33:12_2_1_1"/><b>then </b><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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">G1</font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="measure7.html#T3" target="_self">Th3</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E34:12_2_1_1"/><i><font color="Green">E108</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">GG0</font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <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> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="measure7.html#E2:12"><i><font color="Green">E27</font></i></a>, , <a class="ref" href="supinf_2.html#T59">SUPINF_2:59</a></i>;<br/><a NAME="E35:12_2_1_1"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">F</font> <a href="measure1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G0</font> <a href="measure1.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="measure1.html#K1">.</a> <font color="Olive">m</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_2_1_1_11_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_11_1_1"><b>suppose </b></a><a NAME="E1:12_2_1_1_11_1_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:12_2_1_1_11_1_1"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">G0</font> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="measure7.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E3:12_2_1_1_11_1_1"/>
<font color="Maroon">G1</font> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="measure7.html#E1:12"><i><font color="Green">E25</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:12_2_1_1_11_1_1"/>
<font color="Maroon">F</font> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G0</font> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_11_1_2"><b>suppose </b></a><a NAME="E1:12_2_1_1_11_1_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:12_2_1_1_11_1_2"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">G0</font> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="measure7.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E3:12_2_1_1_11_1_2"/>
<font color="Maroon">G1</font> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="measure7.html#E1:12"><i><font color="Green">E25</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:12_2_1_1_11_1_2"/>
<font color="Maroon">F</font> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G0</font> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E36:12_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">GG0</font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</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"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p4">(<span class="default"><font color="Maroon">G1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">G0</font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">m0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">G1</font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">m0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font></span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="txt" href="measure7.html#E2:12"><i><font color="Green">E27</font></i></a>, , , </i>;<br/><b>hence </b><a NAME="E37:12_2_1_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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">F</font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m0</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m1</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="measure7.html#T2" target="_self">Th2</a>, , , <a class="ref" href="measure1.html#T4">MEASURE1:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_2"><b>suppose </b></a><a NAME="E1:12_2_1_2"/><i><font color="Green">E113</font></i>: 
not <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">m0</font>
;<br/><b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>assume </b><a NAME="E2:12_2_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">F</font> is <a href="supinf_2.html#V6">nonnegative</a>
 ;<br/><a NAME="E3:12_2_1_2"/><i><font color="Green">E115</font></i>: 
<span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p3">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</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/>.= 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/><b>hence </b><a NAME="E5:12_2_1_2"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m0</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:12"/><i><font color="Green">E116</font></i>: 
for <font color="Olive">k</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">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E6:12"/>
for <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">x</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> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> holds <br/> ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</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">G</font></span>)</span> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:12_3"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">x</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> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:12_3"/>
 ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</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">G</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:12_3_1"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">n</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> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">n</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">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:12_3_1"/><i><font color="Green">E119</font></i>: 
for <font color="Olive">F</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>   st <font color="Olive">F</font> is <a href="supinf_2.html#V6">nonnegative</a> holds <br/><span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p3">(<span class="default"><font color="Olive">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">n</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="Olive">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">G</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font>
;<br/>

<a NAME="E6:12_3_1"/>
(  <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">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:12_3_1"/>
( <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">G</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m</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">G</font></span>)</span> &amp; <font color="Maroon">x</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">G</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">m</font> )
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:12_3"/>
 ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</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">G</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">y</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:12"/><b>then </b><i><font color="Green">E120</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> <span class="p3">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</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">G</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T99">SUPINF_1:99</a></i>;<br/>
<a NAME="E8:12"/>
 <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">G</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">G</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#D23">SUPINF_2:def 23</a></i>;<br/>
<b>hence </b><a NAME="E9:12"/>
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><a href="measure7.html#K2" target="_self">On</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="supinf_2.html#D23">SUPINF_2:def 23</a></i>;<br/>


</div>
