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<b>let </b><font color="Maroon">X</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">S</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>









<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:64_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:64_1"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:64_1"/><b>then </b>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  ex <font color="Olive">w</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font> ) )
 <b>by </b><i><a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/><a NAME="E4:64_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:64_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><b>consider </b><font color="Maroon">w</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E7:64_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font> &amp; <font color="Maroon">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font> )
 <b>by </b><i>, <a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/><a NAME="E8:64_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T10" target="_self">Th10</a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E9:64_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T10" target="_self">Th10</a>, <a class="ref" href="mesfunc5.html#T9" target="_self">Th9</a>, <a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:64"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">v</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="E3:64"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span>
 ;<br/>

<a NAME="E4:64"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">w</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E6:64"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font> &amp; <font color="Maroon">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font> )
 <b>by </b><i>, <a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/>
<a NAME="E7:64"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T9" target="_self">Th9</a>, <a class="ref" href="mesfunc5.html#T10" target="_self">Th10</a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E8:64"/><b>then </b>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp;  ex <font color="Olive">w</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font> ) )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T10" target="_self">Th10</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:64"/><b>then </b>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E10:64"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


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