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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">M</font> be    <a href="measure1.html#M3">sigma_Measure</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> , <font color="Maroon">g</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:67"/><i><font color="Green">E35</font></i>: 
 ex <font color="Olive">E1</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">E1</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Olive">E1</font> )
 <b>and </b><br/><a NAME="E2:67"/><i><font color="Green">E37</font></i>: 
 ex <font color="Olive">E2</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">E2</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Olive">E2</font> )
 ;<br/>

<b>consider </b><font color="Maroon">E1</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:67"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">E1</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E1</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">E2</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E6:67"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">E2</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E2</font> )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">E3</font> = <font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font>;<br/>
<b>set </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">g1</font> = <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span>;<br/>
<a NAME="E7:67"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E1</font> &amp; <font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E2</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E8:67"/><b>then </b>
( <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font> &amp; <font color="Maroon">g</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T11" target="_self">Th11</a>, <a class="ref" href="mesfunc5.html#T13" target="_self">Th13</a>, <a class="ref" href="mesfunc1.html#T34">MESFUNC1:34</a></i>;<br/>
<a NAME="E9:67"/><b>then </b><i><font color="Green">E41</font></i>: 
( <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font> &amp; <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E10:67"/>
(  <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> <span class="p2">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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> <span class="p1">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E11:67"/><b>then </b><i><font color="Green">E42</font></i>: 
(  <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> <span class="p2">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T11" target="_self">Th11</a>, <a class="ref" href="mesfunc5.html#T13" target="_self">Th13</a>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<b>then consider </b><font color="Maroon">E</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E13:67"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">E</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#D1" target="_self">Def1</a>, <a class="ref" href="mesfunc5.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E14:67"/><i><font color="Green">E44</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a></i>;<br/>
<a NAME="E15:67"/><i><font color="Green">E47</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T139">FUNCT_1:139</a></i>;<br/>
<b>then </b><i><font color="Green">E94</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span> <a href="measure1.html#K3">/\</a> <span class="p3">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span> <a href="measure1.html#K3">/\</a> <span class="p3">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
;<br/>
<a NAME="E18:67"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E19:67"/><i><font color="Green">E98</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a></i>;<br/>
<a NAME="E20:67"/><b>then </b><i><font color="Green">E99</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T15" target="_self">Th15</a>, <a class="ref" href="mesfunc5.html#D2" target="_self">Def2</a>, , <a class="ref" href="mesfunc5.html#T14" target="_self">Th14</a>, <a class="ref" href="xboole_1.html#T47">XBOOLE_1:47</a>, <a class="ref" href="mesfunc5.html#T11" target="_self">Th11</a>, <a class="ref" href="mesfunc5.html#T13" target="_self">Th13</a></i>;<br/>
<b>take </b>
<font color="Maroon">E</font>
;<br/>

<a NAME="E21:67"/><i><font color="Green">E100</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:67_3"/><i><font color="Green">E101</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"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E</font></span>)</span>
 ;<br/><a NAME="E2:67_3"/><b>then </b>
<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> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E3:67_3"/><b>then </b><i><font color="Green">E102</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="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:67_3"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">E</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D2" target="_self">Def2</a>, , <a class="ref" href="mesfunc5.html#T14" target="_self">Th14</a>, <a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><i><font color="Green">E104</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="mesfunc5.html#D5" target="_self">Def5</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#D6" target="_self">Def6</a>, <a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a></i>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#D4" target="_self">Def4</a>, , <a class="ref" href="mesfunc5.html#D2" target="_self">Def2</a>, <a class="ref" href="mesfunc5.html#D5" target="_self">Def5</a>, <a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T16" target="_self">Th16</a>, , <a class="ref" href="mesfunc5.html#D6" target="_self">Def6</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/><b>hence </b><a NAME="E7:67_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T17" target="_self">Th17</a>, <a class="ref" href="mesfunc5.html#T16" target="_self">Th16</a>, , <a class="ref" href="mesfunc5.html#D6" target="_self">Def6</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/></div><b>end;</b></div>
<a NAME="E23:67"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E1</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">E2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D4" target="_self">Def4</a>, , <a class="ref" href="mesfunc5.html#D2" target="_self">Def2</a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>
<b>hence </b><a NAME="E24:67"/>
( <font color="Maroon">E</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#D3" target="_self">Def3</a>, <a class="ref" href="mesfunc5.html#T15" target="_self">Th15</a>, <a class="ref" href="mesfunc5.html#D2" target="_self">Def2</a>, , <a class="ref" href="mesfunc5.html#T14" target="_self">Th14</a>, <a class="ref" href="xboole_1.html#T47">XBOOLE_1:47</a>, <a class="ref" href="mesfunc5.html#T11" target="_self">Th11</a>, <a class="ref" href="mesfunc5.html#T13" target="_self">Th13</a>, </i>;<br/>


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