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<a NAME="E1:26_1_1"/>
 <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">F0</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span><span class="kw"> such that </span><br/><a NAME="E3:26_1_1"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">and </span><br/><a NAME="E4:26_1_1"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
( <font color="Maroon" title="c3">F0</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c3">F0</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T40" title="MEASURE1:th.40">MEASURE1:40</a></span>;<br/>
<a NAME="E5:26_1_1"/>
(  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c3">F0</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a>, <a class="ref" href="zfmisc_1.html#T93" title="ZFMISC_1:th.93">ZFMISC_1:93</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">F0</font> = <font color="Maroon" title="c3">F0</font> as    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="measure7.html#D2" target="_self" title="MEASURE7:def.2">Def2</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">F</font> being   <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of <font color="Maroon" title="c2">A</font> st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure7.html#K6" title="MEASURE7:func.6">vol</a> <font color="Olive" title="b1">F</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">D</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:26_1_1"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></span>
<a NAME="E9:26_1_1"/>
 <a href="measure7.html#K6" title="MEASURE7:func.6">vol</a> <font color="Maroon" title="c4">F0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26_1_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:26_1_1"/>
not  <a href="measure7.html#K8" title="MEASURE7:func.8">Svc</a> <font color="Maroon" title="c2">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure7.html#D8" target="_self" title="MEASURE7:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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