<?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="measure4.html#M1" target="_self">C_Measure</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:19"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="measure4.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E2:19"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="measure4.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E3:19"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="measure1.html#NM3">N_Sub_set_fam</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> <a href="setfam_1.html#K5">union</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="measure1.html#NM3">N_Sub_set_fam</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:19_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:19_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:19_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="measure4.html#T6" target="_self">Th6</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:19_1_1"/><i><font color="Green">E24</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="supinf_2.html#D14">SUPINF_2:def 14</a></i>;<br/>
<a NAME="E5:19_1_1"/><i><font color="Green">E25</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>7</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:19_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="measure4.html#E4:19_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<b>hence </b><a NAME="E2:19_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure4.html#E1:19_1"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E7:19_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="prob_1.html#K1">.</a> 0 &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>8</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>7</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</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#T5">MEASURE2:5</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E8:19_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="measure4.html#E5:19_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="measure4.html#E7:19_1_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E9:19_1_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:19_1_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:19_1_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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>7</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure4.html#E7:19_1_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E3:19_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="measure4.html#E5:19_1_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:19_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="measure4.html#E1:19_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="measure4.html#E2:19_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="measure4.html#T17" target="_self">Th17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:19_1_1"/><i><font color="Green">E29</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">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="measure4.html#E8:19_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="measure4.html#E9:19_1_1"><i><font color="Green">E28</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E12:19_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="prob_1.html#K1">.</a> 0 &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>9</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>7</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</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#T9">MEASURE2:9</a></i>;<br/>
<a NAME="E13:19_1_1"/><i><font color="Green">E31</font></i>: 
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1">rng</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure4.html#E7:19_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="measure4.html#E12:19_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="measure4.html#T5" target="_self">Th5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E15:19_1_1"/><i><font color="Green">E32</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>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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="measure4.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E16:19_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1">rng</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1">rng</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="measure4.html#E15:19_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="measure4.html#T4" target="_self">Th4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E18:19_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> 0 &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>11</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>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="measure1.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</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#T5">MEASURE2:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <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">c<sub>1</sub></font> ;<br/>
<a NAME="E20:19_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="measure4.html#E2:19"><i><font color="Green">E21</font></i></a>, <a class="ref" href="measure1.html#T54">MEASURE1:54</a></i>;<br/>
<a NAME="E21:19_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="measure4.html#E12:19_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="measure4.html#E15:19_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="measure4.html#E18:19_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E22:19_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="measure4.html#E5:19_1_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E23:19_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> 0 <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> 0
 
<b>by </b><i><a class="txt" href="measure4.html#E21:19_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></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"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E24:19_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<a NAME="E25:19_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> 0 <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E26:19_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> 0</span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T34">XBOOLE_1:34</a></i>;<br/>
<a NAME="E27:19_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="measure4.html#E1:19_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="measure4.html#E4:19_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></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<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></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/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="measure4.html#E18:19_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/>
<a NAME="E29:19_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="txt" href="measure4.html#E22:19_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="measure4.html#E23:19_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="measure4.html#E27:19_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="measure4.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E30:19_1_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:19_1_1_4"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></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"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:19_1_1_4"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</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>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="measure4.html#E18:19_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<font color="Maroon">c<sub>12</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="measure4.html#E15:19_1_1"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="measure4.html#E12:19_1_1"><i><font color="Green">E30</font></i></a></i>
;<br/>
<a NAME="E4:19_1_1_4"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E5:19_1_1_4"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="measure4.html#E5:19_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="measure4.html#E10:19_1_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E6:19_1_1_4"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="measure4.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E7:19_1_1_4"/>
<font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<a NAME="E8:19_1_1_4"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E9:19_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E10:19_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure4.html#E8:19_1_1_4"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:19_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T34">XBOOLE_1:34</a></i>;<br/>
<a NAME="E12:19_1_1_4"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="measure4.html#E1:19_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="measure4.html#E4:19_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E13:19_1_1_4"/>
<font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="measure4.html#E12:19_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="measure4.html#E15:19_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="measure4.html#E18:19_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E14:19_1_1_4"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i><a class="txt" href="measure4.html#E7:19_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E15:19_1_1_4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:19_1_1_4_2"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_4_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:19_1_1_4_2_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>

<i><font color="Green">E51</font></i>: <font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="measure4.html#E18:19_1_1"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K8">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="measure4.html#E15:19_1_1"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="measure4.html#E12:19_1_1"><i><font color="Green">E30</font></i></a></i>
;<br/>
<a NAME="E3:19_1_1_4_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E4:19_1_1_4_2_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E5:19_1_1_4_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E6:19_1_1_4_2_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="measure4.html#E4:19_1_1_4_2_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:19_1_1_4_2_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure4.html#E1:19_1_1_4_2_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="measure4.html#E2:19_1_1_4_2_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<b>hence </b><a NAME="E8:19_1_1_4_2_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="measure4.html#E7:19_1_1"><i><font color="Green">E26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:19_1_1_4_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="measure4.html#E14:19_1_1_4"><i><font color="Green">E48</font></i></a>, <a class="txt" href="measure4.html#E1:19_1_1_4_2"><i><font color="Green">E49</font></i></a>);<br/></i>
<a NAME="E3:19_1_1_4_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 
;<br/>
<a NAME="E4:19_1_1_4_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<b>hence </b><a NAME="E5:19_1_1_4_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="measure4.html#E3:19_1_1_4_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:19_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 
;<br/>
<a NAME="E17:19_1_1_4"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E18:19_1_1_4"/>
<font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></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="E1:19_1_1_4_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<a NAME="E2:19_1_1_4_3"/><b>then </b>
not <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure4.html#E17:19_1_1_4"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:19_1_1_4_3"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="measure4.html#E1:19_1_1_4_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:19_1_1_4"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="prob_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="measure4.html#E12:19_1_1_4"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E20:19_1_1_4"/><i><font color="Green">E51</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &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"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> &amp;  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="measure4.html#E2:19"><i><font color="Green">E21</font></i></a>, <a class="txt" href="measure4.html#E20:19_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a>, <a class="ref" href="supinf_2.html#T59">SUPINF_2:59</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="prob_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>2</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>5</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="prob_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="measure4.html#E4:19_1_1_4"><i><font color="Green">E44</font></i></a>, <a class="txt" href="measure4.html#E6:19_1_1_4"><i><font color="Green">E45</font></i></a>, <a class="txt" href="measure4.html#E19:19_1_1_4"><i><font color="Green">E50</font></i></a>, <a class="ref" href="measure4.html#T14" target="_self">Th14</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="measure4.html#E2:19_1_1_4"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
;<br/>
<b>then </b><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="measure4.html#E1:19_1_1_4"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="measure4.html#E20:19_1_1_4"><i><font color="Green">E51</font></i></a>, <a class="ref" href="measure4.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></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"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="supinf_2.html#T63">SUPINF_2:63</a></i>
;<br/>
<b>hence </b><a NAME="E23:19_1_1_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E31:19_1_1"/><i><font color="Green">E42</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">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="measure4.html#E29:19_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="measure4.html#E30:19_1_1"><i><font color="Green">E41</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E32:19_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="measure4.html#E15:19_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="measure4.html#E18:19_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E33:19_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E34:19_1_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:19_1_1_5"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<i><font color="Green">E46</font></i>: <font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="measure4.html#E18:19_1_1"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K8">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="measure4.html#E15:19_1_1"><i><font color="Green">E32</font></i></a></i>
;<br/>
<a NAME="E3:19_1_1_5"/>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E4:19_1_1_5"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</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="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="measure4.html#E1:19_1_1_5"><i><font color="Green">E45</font></i></a>, <a class="txt" href="measure4.html#E2:19_1_1_5"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<b>hence </b><a NAME="E5:19_1_1_5"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E35:19_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="measure4.html#E33:19_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="measure4.html#E34:19_1_1"><i><font color="Green">E44</font></i></a>);<br/></i>
<a NAME="E36:19_1_1"/><i><font color="Green">E46</font></i>: 
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="measure4.html#E1:19_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="measure4.html#E4:19_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="measure4.html#E13:19_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="measure4.html#E16:19_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E37:19_1_1"/><i><font color="Green">E47</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 <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<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:19_1_1_6"/><i><font color="Green">E48</font></i>: 
<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<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="measure4.html#E31:19_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E2:19_1_1_6"/>
<font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="measure4.html#E35:19_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E3:19_1_1_6"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/>
<b>hence </b><a NAME="E4:19_1_1_6"/>
<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<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="measure4.html#E1:19_1_1_6"><i><font color="Green">E48</font></i></a>, <a class="ref" href="measure4.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E38:19_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  implies <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1_1_7"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/>

<a NAME="E2:19_1_1_7"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="measure4.html#E2:19"><i><font color="Green">E21</font></i></a>, <a class="ref" href="measure1.html#D4">MEASURE1:def 4</a></i>;<br/>
<a NAME="E3:19_1_1_7"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="txt" href="measure4.html#E1:19_1_1_7"><i><font color="Green">E49</font></i></a>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/>
<a NAME="E4:19_1_1_7"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E5:19_1_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="measure4.html#E1:19_1_1_7"><i><font color="Green">E49</font></i></a>, <a class="txt" href="measure4.html#E3:19_1_1_7"><i><font color="Green">E50</font></i></a>, <a class="ref" href="measure4.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E39:19_1_1"/>
( not <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  implies <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1_1_8"/><i><font color="Green">E49</font></i>: 
( not <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/>

<a NAME="E2:19_1_1_8"/><i><font color="Green">E50</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 <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:19_1_1_8_1"/>
<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<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure4.html#E37:19_1_1"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:19_1_1_8_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="measure4.html#E1:19_1_1_8"><i><font color="Green">E49</font></i></a>, <a class="ref" href="measure4.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 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="E3:19_1_1_8"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K6">ExtREAL</a>  &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_8_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:19_1_1_8_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:19_1_1_8_2"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K6">ExtREAL</a>  &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>17</sub></font>,<font color="Olive">b<sub>1</sub></font>] ) )
 
;<br/>
<b>hence </b><a NAME="E4:19_1_1_8_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K6">ExtREAL</a>  &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>16</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:19_1_1_8"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a>  st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="supinf_2.html#K13">.</a> <font color="Olive">b<sub>2</sub></font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="measure4.html#E3:19_1_1_8"><i><font color="Green">E51</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E6:19_1_1_8"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="supinf_2.html#K13">.</a> <font color="Olive">b<sub>1</sub></font>]
 ;<br/>
<a NAME="E7:19_1_1_8"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> <a href="measure1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_8_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_8_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_1_8_3_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_8_3_1_1"><b>suppose </b></a><a NAME="E1:19_1_1_8_3_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:19_1_1_8_3_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E3:19_1_1_8_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="measure4.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E4:19_1_1_8_3_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="measure4.html#E1:19_1_1_8"><i><font color="Green">E49</font></i></a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/><a NAME="E5:19_1_1_8_3_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#D4">XXREAL_0:def 4</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:19_1_1_8_3_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="measure4.html#E1:19_1_1_8"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E8:19_1_1_8_3_1_1"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="measure4.html#E7:19_1_1_8_3_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/><b>thus </b><a NAME="E9:19_1_1_8_3_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="measure4.html#E6:19_1_1_8"><i><font color="Green">E52</font></i></a>, <a class="txt" href="measure4.html#E1:19_1_1_8_3_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="measure4.html#E4:19_1_1_8_3_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="measure4.html#E8:19_1_1_8_3_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_8_3_1_2"><b>suppose </b></a><a NAME="E1:19_1_1_8_3_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_1_1_8_3_1_2"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="measure4.html#E6:19_1_1_8"><i><font color="Green">E52</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E8:19_1_1_8"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>16</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T58">SUPINF_2:58</a></i>;<br/>
<a NAME="E9:19_1_1_8"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">b<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="Maroon">c<sub>16</sub></font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_8_4"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>;<br/>
<b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:19_1_1_8_4"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="measure1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <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="E2:19_1_1_8_4"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[0]
 
<b>by </b><i><a class="txt" href="measure4.html#E6:19_1_1_8"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E3:19_1_1_8_4"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_8_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:19_1_1_8_4_1"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></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> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="supinf_2.html#T63">SUPINF_2:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i><a class="txt" href="measure4.html#E6:19_1_1_8"><i><font color="Green">E52</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="measure4.html#E1:19_1_1_8_4_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:19_1_1_8_4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="measure4.html#E2:19_1_1_8_4"><i><font color="Green">E55</font></i></a>, <a class="txt" href="measure4.html#E3:19_1_1_8_4"><i><font color="Green">E56</font></i></a>);<br/></i>
<a NAME="E5:19_1_1_8_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E6:19_1_1_8_4"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="measure4.html#E2:19_1_1_8"><i><font color="Green">E50</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_8_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:19_1_1_8_5"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/><a NAME="E2:19_1_1_8_5"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><b>hence </b><a NAME="E3:19_1_1_8_5"/>
<font color="Maroon">c<sub>16</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="measure4.html#E6:19_1_1_8"><i><font color="Green">E52</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:19_1_1_8"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="supinf_2.html#K19">SUM</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="measure1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="measure4.html#E8:19_1_1_8"><i><font color="Green">E53</font></i></a>, <a class="ref" href="supinf_2.html#T67">SUPINF_2:67</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="measure1.html#K1">.</a> 0;<br/>
<a NAME="E12:19_1_1_8"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="measure1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <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="E13:19_1_1_8"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="measure1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure4.html#E6:19_1_1_8"><i><font color="Green">E52</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="measure1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="measure1.html#K1">.</a> 0</span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <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"><span class="p2">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="measure1.html#K1">.</a> 0</span>)</span> <a href="supinf_2.html#K2">+</a> <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i><a class="txt" href="measure4.html#E6:19_1_1_8"><i><font color="Green">E52</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="measure4.html#E13:19_1_1_8"><i><font color="Green">E56</font></i></a>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/>
<a NAME="E15:19_1_1_8"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure4.html#E9:19_1_1_8"><i><font color="Green">E54</font></i></a>, <a class="txt" href="measure4.html#E11:19_1_1_8"><i><font color="Green">E55</font></i></a>, <a class="ref" href="measure3.html#T3">MEASURE3:3</a></i>;<br/>
<a NAME="E16:19_1_1_8"/>
<font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure4.html#E36:19_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="measure4.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E17:19_1_1_8"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure4.html#E15:19_1_1_8"><i><font color="Green">E57</font></i></a>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/>
<b>hence </b><a NAME="E18:19_1_1_8"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="measure4.html#E1:19_1_1_8"><i><font color="Green">E49</font></i></a>, <a class="ref" href="measure4.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E40:19_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="measure4.html#E2:19"><i><font color="Green">E21</font></i></a>, <a class="txt" href="measure4.html#E38:19_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="measure1.html#D4">MEASURE1:def 4</a>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:19_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="measure4.html#D3" target="_self">Def3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="prob_1.html#V1">compl-closed</a> <a href="measure1.html#V2">sigma-additive</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="measure4.html#E1:19"><i><font color="Green">E20</font></i></a>, <a class="ref" href="measure1.html#D3">MEASURE1:def 3</a>, <a class="ref" href="measure1.html#D9">MEASURE1:def 9</a></i>;<br/>
<a NAME="E5:19"/>
<font color="Maroon">c<sub>3</sub></font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>hence </b><a NAME="E6:19"/>
 <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">c<sub>2</sub></font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div>
