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<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">S</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">M</font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</font>;<br/><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> ,<font color="Maroon">S</font>, <font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">S</font>;<br/>







<b>assume </b><a NAME="E1:11"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</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>  &amp; <font color="Maroon">G</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">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</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">F</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">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> ) ) )
 ;<br/>

<a NAME="E2:11"/><b>then </b><i><font color="Green">E26</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">G</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</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">F</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:11"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</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">G</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</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">F</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:11"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</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">F</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="11_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</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">F</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/>

<a NAME="E2:11_1"/>
 <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">G</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<b>hence </b><a NAME="E3:11_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:11"/>
<font color="Maroon">M</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">G</font></span>)</span></span>)</span> <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="E6:11"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">M</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">G</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i>, , <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/>
<a NAME="E7:11"/>
 <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">G</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<b>hence </b><a NAME="E8:11"/>
<font color="Maroon">M</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">F</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</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">M</font> <a href="measure1.html#K1">.</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">G</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="measure1.html#T63">MEASURE1:63</a></i>;<br/>


</div>
