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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">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="supinf_1.html#K3">REAL</a> ;<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">B</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font>;<br/>









<b>assume </b><a NAME="E1:88"/><i><font color="Green">E37</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 ;<br/>

<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>now </b></a><div class="add"><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="88_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:88_1_1"/>
( <font color="Maroon">x</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="mesfunc6.html#K1" target="_self">less_dom</a> <font color="Maroon">f</font>,<font color="Maroon">r</font></span>)</span> iff ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc6.html#K1" target="_self">less_dom</a> <font color="Maroon">f</font>,<font color="Maroon">r</font> ) )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E2:88_1_1"/>
( <font color="Maroon">x</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="mesfunc6.html#K1" target="_self">less_dom</a> <font color="Maroon">f</font>,<font color="Maroon">r</font></span>)</span> iff <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc6.html#K1" target="_self">less_dom</a> <font color="Maroon">f</font>,<font color="Maroon">r</font> )
 <b>by </b><i>, <a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/></div><b>end;</b></div><a NAME="E2:88_1"/><b>then </b>
( <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" target="_self">less_dom</a> <font color="Maroon">f</font>,<font color="Maroon">r</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="mesfunc6.html#K1" target="_self">less_dom</a> <font color="Maroon">f</font>,<font color="Maroon">r</font> &amp;  <a href="mesfunc6.html#K1" target="_self">less_dom</a> <font color="Maroon">f</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" target="_self">less_dom</a> <font color="Maroon">f</font>,<font color="Maroon">r</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E3:88_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" target="_self">less_dom</a> <font color="Maroon">f</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="mesfunc6.html#K1" target="_self">less_dom</a> <font color="Maroon">f</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:88_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc6.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">B</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2_1"><b>now </b></a><div class="add"><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/><span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" target="_self">less_dom</a> <font color="Maroon">f</font>,<font color="Maroon">r</font></span>)</span> = 
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc6.html#K1" target="_self">less_dom</a> <font color="Maroon">f</font>,<font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" target="_self">less_dom</a> <font color="Maroon">f</font>,<font color="Maroon">r</font></span>)</span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E2:88_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" target="_self">less_dom</a> <font color="Maroon">f</font>,<font color="Maroon">r</font></span>)</span> <a href="mesfunc1.html#R1">is_measurable_on</a> <font color="Maroon">S</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:88_2"/>
<font color="Maroon">f</font> <a href="mesfunc6.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">A</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:88"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc6.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">A</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">B</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_3"><b>now </b></a><div class="add"><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/><span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" target="_self">less_dom</a> <font color="Maroon">f</font>,<font color="Maroon">r</font></span>)</span> = 
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc6.html#K1" target="_self">less_dom</a> <font color="Maroon">f</font>,<font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" target="_self">less_dom</a> <font color="Maroon">f</font>,<font color="Maroon">r</font></span>)</span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E2:88_3"/>
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" target="_self">less_dom</a> <font color="Maroon">f</font>,<font color="Maroon">r</font></span>)</span> <a href="mesfunc1.html#R1">is_measurable_on</a> <font color="Maroon">S</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:88"/>
<font color="Maroon">f</font> <a href="mesfunc6.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/>


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