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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="measure7.html#M1" target="_self">Interval_Covering</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure7.html#K7" target="_self">vol</a> <font color="Olive">b<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:26_1_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff ( <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>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon">c<sub>4</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> <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E4:26_1_1"/><i><font color="Green">E25</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="supinf_1.html#K3">REAL</a> ,<a href="measure5.html#K6">{}</a> </span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a>  ) )
 <b>by </b><i><a class="txt" href="measure7.html#E14"><i><font color="Green">Lemma13</font></i></a>, <a class="ref" href="measure1.html#T40">MEASURE1:40</a></i>;<br/>
<a NAME="E5:26_1_1"/><i><font color="Green">E26</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="supinf_1.html#K3">REAL</a> ,<a href="measure5.html#K6">{}</a> </span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <a href="supinf_1.html#K3">REAL</a>  <a href="xboole_0.html#K2">\/</a> <a href="measure5.html#K6">{}</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T93">ZFMISC_1:93</a></i>;<br/>
<a NAME="E6:26_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is   <a href="measure5.html#NM1">Interval</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:26_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<b>hence </b><a NAME="E2:26_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is   <a href="measure5.html#NM1">Interval</a>
 <b>by </b><i><a class="txt" href="measure7.html#E4:26_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="measure6.html#T35">MEASURE6:35</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="measure7.html#E4:26_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="measure7.html#E5:26_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="measure7.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E8:26_1_1"/>
 <a href="measure7.html#K7" target="_self">vol</a> <font color="Maroon">c<sub>5</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="measure7.html#E2:26_1_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:26_1_1"/>
not  <a href="measure7.html#K9" target="_self">Svc</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="measure7.html#D8" target="_self">Def8</a></i>;<br/>


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