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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:17"/><i><font color="Green">E11</font></i>: 
( ( 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">c<sub>4</sub></font> <a href="funct_2.html#K3">.</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> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>3</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:17"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &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">c<sub>5</sub></font> <a href="funct_2.html#K3">.</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="measure2.html#T10">MEASURE2:10</a></i>;<br/>
<a NAME="E4:17"/>
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">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</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>by </b><i><a class="txt" href="measure3.html#E1:17"><i><font color="Green">E11</font></i></a>, <a class="txt" href="measure3.html#E3:17"><i><font color="Green">E12</font></i></a>, <a class="ref" href="measure2.html#T15">MEASURE2:15</a></i>;<br/>
<a NAME="E5:17"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <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"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="measure2.html#T27">MEASURE2:27</a></i>;<br/>
<a NAME="E6:17"/><i><font color="Green">E14</font></i>: 
 <a href="measure2.html#K2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K1">meet</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="measure3.html#E1:17"><i><font color="Green">E11</font></i></a>, <a class="txt" href="measure3.html#E3:17"><i><font color="Green">E12</font></i></a>, <a class="ref" href="measure3.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E7:17"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><a href="measure2.html#K2">union</a> <span class="p3">(<span class="default"><a href="measure3.html#K1" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K1">meet</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="measure3.html#E1:17"><i><font color="Green">E11</font></i></a>, <a class="txt" href="measure3.html#E3:17"><i><font color="Green">E12</font></i></a>, <a class="ref" href="measure3.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E8:17"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure2.html#K2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K1">meet</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="measure1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure2.html#K2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:17_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><a href="measure2.html#K1">meet</a> <span class="p3">(<span class="default"><a href="measure3.html#K1" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><a href="measure2.html#K1">meet</a> <span class="p3">(<span class="default"><a href="measure3.html#K1" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/>

<a NAME="E2:17_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> 0
 
<b>by </b><i><a class="txt" href="measure3.html#E6:17"><i><font color="Green">E14</font></i></a>, <a class="txt" href="measure3.html#E1:17_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<b>hence </b><a NAME="E3:17_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure3.html#E1:17"><i><font color="Green">E11</font></i></a>, <a class="txt" href="measure3.html#E6:17"><i><font color="Green">E14</font></i></a>, <a class="txt" href="measure3.html#E1:17_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="measure3.html#E1:17_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:17_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/>
<a NAME="E4:17_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="txt" href="measure3.html#E6:17"><i><font color="Green">E14</font></i></a>, <a class="txt" href="measure3.html#E1:17_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="measure3.html#E2:17_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/>
<a NAME="E5:17_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> 0
 
<b>by </b><i><a class="txt" href="measure3.html#E6:17"><i><font color="Green">E14</font></i></a>, <a class="txt" href="measure3.html#E1:17_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<b>hence </b><a NAME="E6:17_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K1">meet</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="measure3.html#E7:17"><i><font color="Green">E15</font></i></a>, <a class="txt" href="measure3.html#E1:17_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="measure3.html#E4:17_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="measure1.html#T63">MEASURE1:63</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:17"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K1">meet</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="measure3.html#E5:17"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:17"/>
<font color="Maroon">c<sub>3</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K1">meet</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="measure3.html#E1:17"><i><font color="Green">E11</font></i></a>, <a class="txt" href="measure3.html#E3:17"><i><font color="Green">E12</font></i></a>, <a class="ref" href="measure3.html#T13" target="_self">Th13</a></i>;<br/>


</div>
