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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">C</font> be    <a href="measure4.html#M1" target="_self">C_Measure</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">W</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">Z</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>





<b>reconsider </b><font color="Maroon">P</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:11"/><i><font color="Green">E23</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">W</font>,<font color="Maroon">Z</font>,<font color="Maroon">P</font></span><a href="enumset1.html#K1">}</a></span> &amp; <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> &amp; <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">n</font> holds <br/><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> ) )
 <b>by </b><i><a class="ref" href="measure1.html#T38">MEASURE1:38</a></i>;<br/>
<a NAME="E4:11"/><i><font color="Green">E24</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">W</font>,<font color="Maroon">Z</font>,<font color="Maroon">P</font></span><a href="enumset1.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:11_1"/>
 <a href="tarski.html#K3">union</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">W</font>,<font color="Maroon">Z</font>,<font color="Maroon">P</font></span><a href="enumset1.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Z</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:11_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">W</font>,<font color="Maroon">Z</font>,<font color="Maroon">P</font></span><a href="enumset1.html#K1">}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:11_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">W</font>,<font color="Maroon">Z</font>,<font color="Maroon">P</font></span><a href="enumset1.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:11_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 
<b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:11_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:11_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Z</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_2_1_1"><b>suppose </b></a><a NAME="E1:11_1_2_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">Y</font> = <font color="Maroon">W</font>;<br/><b>thus </b><a NAME="E2:11_1_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">W</font>,<font color="Maroon">Z</font>,<font color="Maroon">P</font></span><a href="enumset1.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_2_1_2"><b>suppose </b></a><a NAME="E1:11_1_2_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">Y</font> = <font color="Maroon">Z</font>;<br/><b>thus </b><a NAME="E2:11_1_2_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">W</font>,<font color="Maroon">Z</font>,<font color="Maroon">P</font></span><a href="enumset1.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:11_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">W</font>,<font color="Maroon">Z</font>,<font color="Maroon">P</font></span><a href="enumset1.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:11"/>
<font color="Maroon">C</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i/>;<br/>
<a NAME="E6:11"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</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="E7:11"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">r</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="11_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:11_2"/>
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:11_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
;<br/>
<a NAME="E3:11_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:11_2"/><b>then </b>
<font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:11_2"/><b>then </b>
<font color="Maroon">C</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:11_2"/>
<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:11"/>
( <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> &amp; <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E9:11"/><b>then </b><i><font color="Green">E37</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="measure1.html#K1">.</a> <font color="Maroon">P</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="measure1.html#K1">.</a> <font color="Maroon">Z</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="measure1.html#K1">.</a> <font color="Maroon">W</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<b>set </b><font color="Maroon">G</font> = <font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font>;<br/>
<b>consider </b><font color="Maroon">y1</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">y2</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E11:11"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">y1</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"><font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 1 &amp; <font color="Maroon">y2</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"><font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0 )
 ;<br/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 2 = 
<font color="Maroon">y1</font> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default">1 <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"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 1</span>)</span> <a href="supinf_2.html#K2">+</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"><font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
<br/>.= 
<font color="Maroon">y2</font> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default">0 <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"><font color="Maroon">C</font> <a href="measure1.html#K1">.</a> <font color="Maroon">W</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="measure1.html#K1">.</a> <font color="Maroon">Z</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="supinf_2.html#T63">SUPINF_2:63</a></i>
;<br/>
<a NAME="E13:11"/><b>then </b>
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="measure1.html#K1">.</a> <font color="Maroon">W</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="measure1.html#K1">.</a> <font color="Maroon">Z</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="supinf_2.html#T67">SUPINF_2:67</a></i>;<br/>
<b>hence </b><a NAME="E14:11"/>
<font color="Maroon">C</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Z</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="measure1.html#K1">.</a> <font color="Maroon">W</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="measure1.html#K1">.</a> <font color="Maroon">Z</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div>
