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<div class="add">

<b>let </b><font color="Maroon">C</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">C</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/>



<b>thus </b><a NAME="E1:17"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">1 <a href="mesfunc1.html#K11" target="_self">/</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_2.html#K1">0.</a> </span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E2:17"/>
for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>  st <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">1 <a href="mesfunc1.html#K11" target="_self">/</a> <font color="Maroon">f</font></span>)</span> holds <br/><span class="p1">(<span class="default">1 <a href="mesfunc1.html#K11" target="_self">/</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <a href="mesfunc1.html#K10" target="_self">1.</a>  <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Olive">c</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:17_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">1 <a href="mesfunc1.html#K11" target="_self">/</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<a NAME="E2:17_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="mesfunc1.html#K11" target="_self">/</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 1</span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">c</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:17_1"/>
<span class="p1">(<span class="default">1 <a href="mesfunc1.html#K11" target="_self">/</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <a href="mesfunc1.html#K10" target="_self">1.</a>  <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">c</font></span>)</span>
 ;<br/>


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


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