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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">M</font> be    <a href="measure1.html#M3" title="MEASURE1:mode.3">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/>







<b>assume </b><a NAME="E1:143"/><i><font color="Green" title="E128">A1</font></i>: 
(  ex <font color="Olive" title="b1">E0</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">E0</font> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Olive" title="b1">E0</font> ) &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E3:143"/><i><font color="Green" title="E129">A2</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c6">E</font> )
 ;<br/>
<a NAME="E4:143"/><i><font color="Green" title="E130">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D3" title="MESFUNC1:def.3">MESFUNC1:def 3</a></i>;<br/>
<a NAME="E5:143"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E6:143"/><b>then </b><i><font color="Green" title="E131">A4</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T18" title="XBOOLE_1:th.18">XBOOLE_1:18</a></i>;<br/>
<a NAME="E7:143"/><b>then </b><i><font color="Green" title="E132">A5</font></i>: 
( <font color="Maroon" title="c6">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> &amp; <font color="Maroon" title="c6">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> )
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a>, <a class="txt" href="mesfunc5.html#E3:143"><i><font color="Green" title="E129">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">E1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E9:143"/><i><font color="Green" title="E133">A6</font></i>: 
( <font color="Maroon" title="c7">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c7">E1</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#D17" target="_self" title="MESFUNC5:def.17">Def17</a>, <a class="txt" href="mesfunc5.html#E1:143"><i><font color="Green" title="E128">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">E2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E11:143"/><i><font color="Green" title="E134">A7</font></i>: 
( <font color="Maroon" title="c8">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font> &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c8">E2</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#D17" target="_self" title="MESFUNC5:def.17">Def17</a>, <a class="txt" href="mesfunc5.html#E1:143"><i><font color="Green" title="E128">A1</font></i></a></i>;<br/>
<a NAME="E12:143"/>
( <span class="p1"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp; <span class="p1"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:143"><i><font color="Green" title="E128">A1</font></i></a>, <a class="txt" href="mesfunc5.html#E9:143"><i><font color="Green" title="E133">A6</font></i></a>, <a class="txt" href="mesfunc5.html#E11:143"><i><font color="Green" title="E134">A7</font></i></a>, <a class="ref" href="mesfunc5.html#T106" target="_self" title="MESFUNC5:th.106">Th106</a></i>;<br/>
<a NAME="E13:143"/><b>then </b><i><font color="Green" title="E135">A8</font></i>: 
( <span class="p1"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp; <span class="p1"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T103" target="_self" title="MESFUNC5:th.103">Th103</a></i>;<br/>
<a NAME="E14:143"/>
( <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c6">E</font> &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c6">E</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E9:143"><i><font color="Green" title="E133">A6</font></i></a>, <a class="txt" href="mesfunc5.html#E11:143"><i><font color="Green" title="E134">A7</font></i></a>, <a class="txt" href="mesfunc5.html#E3:143"><i><font color="Green" title="E129">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E6:143"><i><font color="Green" title="E131">A4</font></i></a>, <a class="ref" href="mesfunc1.html#T34" title="MESFUNC1:th.34">MESFUNC1:34</a></i>;<br/>
<a NAME="E15:143"/><b>then </b><i><font color="Green" title="E136">A9</font></i>: 
( <span class="p1"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c6">E</font> &amp; <span class="p1"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c6">E</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:143"><i><font color="Green" title="E129">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E6:143"><i><font color="Green" title="E131">A4</font></i></a>, <a class="ref" href="mesfunc2.html#T29" title="MESFUNC2:th.29">MESFUNC2:29</a></i>;<br/>
<a NAME="E16:143"/>
( <span class="p1"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <span class="p1"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E35"><i><font color="Green" title="E31">Lm1</font></i></a></i>;<br/>
<a NAME="E17:143"/><b>then </b><i><font color="Green" title="E137">A10</font></i>: 
( <span class="p1"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <span class="p1"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T21" target="_self" title="MESFUNC5:th.21">Th21</a></i>;<br/>
<a NAME="E18:143"/><b>then </b><i><font color="Green" title="E138">A11</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E13:143"><i><font color="Green" title="E135">A8</font></i></a>, <a class="ref" href="mesfunc5.html#T112" target="_self" title="MESFUNC5:th.112">Th112</a></i>;<br/>
<a NAME="E19:143"/>
( <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font> &amp; <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:143"><i><font color="Green" title="E132">A5</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E20:143"/><b>then </b><i><font color="Green" title="E139">A12</font></i>: 
( <font color="Maroon" title="c6">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> &amp; <span class="p1"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c6">E</font> &amp; <font color="Maroon" title="c6">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> &amp; <span class="p1"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c6">E</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E15:143"><i><font color="Green" title="E136">A9</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a>, <a class="ref" href="mesfunc5.html#T48" target="_self" title="MESFUNC5:th.48">Th48</a></i>;<br/>
<a NAME="E21:143"/><b>then </b><i><font color="Green" title="E140">A13</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c6">E</font>
 
;<br/>
<a NAME="E22:143"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">ff</font> = <font color="Maroon" title="c4">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></i>;<br/>
<a NAME="E24:143"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font> holds <br/><font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">gg</font> = <font color="Maroon" title="c5">g</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></i>;<br/>
<div><i><font color="Green" title="E141">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="143_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:143_1"/><i><font color="Green" title="E142">A15</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span>
 ;<br/><a NAME="E2:143_1"/><b>then </b>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:143"><i><font color="Green" title="E129">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E7:143"><i><font color="Green" title="E132">A5</font></i></a></i>;<br/><a NAME="E3:143_1"/><b>then </b><i><font color="Green" title="E143">A16</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">ff</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">gg</font> )
 <b>by </b><i><a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/><a NAME="E4:143_1"/><i><font color="Green" title="E144">A17</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:143_1"><i><font color="Green" title="E142">A15</font></i></a>, <a class="txt" href="mesfunc5.html#E3:143"><i><font color="Green" title="E129">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E17:143"><i><font color="Green" title="E137">A10</font></i></a>, <a class="txt" href="mesfunc5.html#E21:143"><i><font color="Green" title="E140">A13</font></i></a>, <a class="ref" href="mesfunc5.html#T28" target="_self" title="MESFUNC5:th.28">Th28</a></i>;<br/><a NAME="E5:143_1"/>
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="143_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:143_1_1"/><i><font color="Green" title="E145">A18</font></i>: 
( ( <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) or ( <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="143_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:143_1_1_1"/>
( ( <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) or ( <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) )
 </span><b>by </b><i><a class="txt" href="mesfunc5.html#E1:143_1_1"><i><font color="Green" title="E145">A18</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="143_1_1_1_1"><b>suppose </b></a><a NAME="E1:143_1_1_1_1"/>
( <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 ;<br/><div class="add"><a NAME="E2:143_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c9">ff</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c10">gg</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:143_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ff</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">gg</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:143_1"><i><font color="Green" title="E143">A16</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/><a NAME="E4:143_1_1_1_1"/><b>then </b><i><font color="Green" title="E146">A19</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ff</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">gg</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:143_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">ff</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">gg</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">ff</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">gg</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">ff</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">gg</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E6:143_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:143_1"><i><font color="Green" title="E142">A15</font></i></a>, <a class="txt" href="mesfunc5.html#E4:143"><i><font color="Green" title="E130">A3</font></i></a>, <a class="txt" href="mesfunc5.html#E4:143_1_1_1_1"><i><font color="Green" title="E146">A19</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="143_1_1_1_2"><b>suppose </b></a><a NAME="E1:143_1_1_1_2"/>
( <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 ;<br/><div class="add"><a NAME="E2:143_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c9">ff</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c10">gg</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:143_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ff</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">gg</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:143_1"><i><font color="Green" title="E143">A16</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/><a NAME="E4:143_1_1_1_2"/><b>then </b><i><font color="Green" title="E146">A20</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ff</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">gg</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:143_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">ff</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">gg</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">ff</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">gg</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">ff</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">gg</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E6:143_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:143_1"><i><font color="Green" title="E142">A15</font></i></a>, <a class="txt" href="mesfunc5.html#E4:143"><i><font color="Green" title="E130">A3</font></i></a>, <a class="txt" href="mesfunc5.html#E4:143_1_1_1_2"><i><font color="Green" title="E146">A20</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E6:143_1"/><b>then </b><i><font color="Green" title="E145">A21</font></i>: 
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 <b>by </b><i><a class="ref" href="extreal2.html#T61" title="EXTREAL2:th.61">EXTREAL2:61</a></i>;<br/><span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
<b>by </b><i><a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a>, <a class="txt" href="mesfunc5.html#E3:143"><i><font color="Green" title="E129">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E1:143_1"><i><font color="Green" title="E142">A15</font></i></a>, <a class="txt" href="mesfunc5.html#E7:143"><i><font color="Green" title="E132">A5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a>, <a class="txt" href="mesfunc5.html#E3:143"><i><font color="Green" title="E129">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E1:143_1"><i><font color="Green" title="E142">A15</font></i></a>, <a class="txt" href="mesfunc5.html#E7:143"><i><font color="Green" title="E132">A5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:143_1"><i><font color="Green" title="E142">A15</font></i></a>, <a class="txt" href="mesfunc5.html#E3:143"><i><font color="Green" title="E129">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E20:143"><i><font color="Green" title="E139">A12</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:143_1"><i><font color="Green" title="E142">A15</font></i></a>, <a class="txt" href="mesfunc5.html#E3:143"><i><font color="Green" title="E129">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E20:143"><i><font color="Green" title="E139">A12</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:143_1"><i><font color="Green" title="E144">A17</font></i></a>, <a class="ref" href="mesfunc1.html#D3" title="MESFUNC1:def.3">MESFUNC1:def 3</a></i>
;<br/><b>hence </b><a NAME="E8:143_1"/>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:143_1"><i><font color="Green" title="E142">A15</font></i></a>, <a class="ref" href="mesfunc1.html#D3" title="MESFUNC1:def.3">MESFUNC1:def 3</a>, <a class="txt" href="mesfunc5.html#E6:143_1"><i><font color="Green" title="E145">A21</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E27:143"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E21:143"><i><font color="Green" title="E140">A13</font></i></a>, <a class="ref" href="mesfunc5.html#T28" target="_self" title="MESFUNC5:th.28">Th28</a>, <a class="txt" href="mesfunc5.html#E17:143"><i><font color="Green" title="E137">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E28:143"/>
<font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">+</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E18:143"><i><font color="Green" title="E138">A11</font></i></a>, <a class="txt" href="mesfunc5.html#E26:143"><i><font color="Green" title="E141">A14</font></i></a>, <a class="txt" href="mesfunc5.html#E3:143"><i><font color="Green" title="E129">A2</font></i></a>, <a class="ref" href="mesfunc5.html#T108" target="_self" title="MESFUNC5:th.108">Th108</a></i>;<br/>


</div>
