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<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st  ex <font color="Olive" title="b5">E0</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font> st <br/>(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <font color="Olive" title="b4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">E0</font> &amp; <font color="Olive" title="b3">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <font color="Olive" title="b4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b5">E0</font> ) &amp; <font color="Olive" title="b3">f</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b2">M</font> &amp; <font color="Olive" title="b4">g</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b2">M</font> holds <br/><font color="Olive" title="b3">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <font color="Olive" title="b4">g</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b2">M</font></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st  ex <font color="Olive" title="b4">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#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <font color="Olive" title="b3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">E0</font> &amp; <font color="Olive" title="b2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <font color="Olive" title="b3">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b4">E0</font> ) &amp; <font color="Olive" title="b2">f</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b3">g</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b1">M</font> holds <br/><font color="Olive" title="b2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <font color="Olive" title="b3">g</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b1">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st  ex <font color="Olive" title="b3">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#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">E0</font> &amp; <font color="Olive" title="b1">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <font color="Olive" title="b2">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b3">E0</font> ) &amp; <font color="Olive" title="b1">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="Olive" title="b2">g</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> holds <br/><font color="Olive" title="b1">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <font color="Olive" title="b2">g</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font></span><br/><span class="kw">let </span><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#K7" title="NUMBERS:func.7">ExtREAL</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  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#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</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#K4" title="MESFUNC1:func.4">+</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">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> implies <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</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> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:136"/><span class="lab"><font color="Green" title="E122">A1</font></span>: 
 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#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</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#K4" title="MESFUNC1:func.4">+</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E0</font> )
 <span class="kw">and </span><br/><a NAME="E2:136"/><span class="lab"><font color="Green" title="E123">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:136"/><span class="lab"><font color="Green" title="E124">A3</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</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></span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:136"/><span class="lab"><font color="Green" title="E125">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</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>
 <span class="kw">and </span><br/><a NAME="E6:136"/><span class="lab"><font color="Green" title="E126">A5</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c6">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136"><span class="lab"><font color="Green" title="E122">A1</font></span></a></span>;<br/>
<a NAME="E7:136"/><span class="lab"><font color="Green" title="E127">A6</font></span>: 
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="mesfunc5.html#E29"><span class="lab"><font color="Green" title="E26">Lm1</font></span></a>, <a class="ref" href="mesfunc5.html#T15" target="_self" title="MESFUNC5:th.15">Th21</a></span>;<br/>
<a NAME="E8:136"/>
 ex <font color="Olive" title="b1">E2</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/>( <font color="Olive" title="b1">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:136"><span class="lab"><font color="Green" title="E124">A3</font></span></a>, <a class="ref" href="mesfunc5.html#D17" target="_self" title="MESFUNC5:def.17">Def17</a></span>;<br/>
<a NAME="E9:136"/><span class="kw">then </span>
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:136"><span class="lab"><font color="Green" title="E124">A3</font></span></a>, <a class="ref" href="mesfunc5.html#T100" target="_self" title="MESFUNC5:th.100">Th106</a></span>;<br/>
<a NAME="E10:136"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A7</font></span>: 
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#T97" target="_self" title="MESFUNC5:th.97">Th103</a></span>;<br/>
<a NAME="E11:136"/><span class="lab"><font color="Green" title="E129">A8</font></span>: 
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="mesfunc5.html#E29"><span class="lab"><font color="Green" title="E26">Lm1</font></span></a>, <a class="ref" href="mesfunc5.html#T15" target="_self" title="MESFUNC5:th.15">Th21</a></span>;<br/>
<a NAME="E12:136"/><span class="lab"><font color="Green" title="E130">A9</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</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#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">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#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D3" title="MESFUNC1:def.3">MESFUNC1:def 3</a></span>;<br/>
<a NAME="E13:136"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T18" title="XBOOLE_1:th.18">XBOOLE_1:18</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E14:136"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A10</font></span>: 
<font color="Maroon" title="c6">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:136"><span class="lab"><font color="Green" title="E125">A4</font></span></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></span>;<br/>
<a NAME="E15:136"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A11</font></span>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E16:136"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:136"><span class="lab"><font color="Green" title="E130">A9</font></span></a>, <a class="ref" href="xboole_1.html#T18" title="XBOOLE_1:th.18">XBOOLE_1:18</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E17:136"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A12</font></span>: 
<font color="Maroon" title="c6">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:136"><span class="lab"><font color="Green" title="E125">A4</font></span></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></span>;<br/>
<a NAME="E18:136"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E19:136"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A13</font></span>: 
<font color="Maroon" title="c6">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/>
<a NAME="E20:136"/><span class="kw">then </span><span class="lab"><font color="Green" title="E135">A14</font></span>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</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#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</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="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Maroon" title="c6">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:136"><span class="lab"><font color="Green" title="E132">A11</font></span></a>, <a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/>
<a NAME="E21:136"/><span class="kw">then </span><span class="lab"><font color="Green" title="E136">A15</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</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> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:136"><span class="lab"><font color="Green" title="E127">A6</font></span></a>, <a class="txt" href="#E11:136"><span class="lab"><font color="Green" title="E129">A8</font></span></a>, <a class="ref" href="mesfunc5.html#T22" target="_self" title="MESFUNC5:th.22">Th28</a></span>;<br/>
<a NAME="E22:136"/><span class="lab"><font color="Green" title="E137">A16</font></span>: 
<font color="Maroon" title="c6">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:136"><span class="lab"><font color="Green" title="E132">A11</font></span></a>, <a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E138">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <font color="Maroon" title="c5">g</font></span>)</span> implies <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</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="c5">g</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</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#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</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> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</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="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font> )</span><br/><a NAME="E1:136_1"/><span class="lab"><font color="Green" title="E139">A18</font></span>: 
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</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="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="extreal2.html#T13" title="EXTREAL2:th.13">EXTREAL2:13</a></span>;<br/><span class="kw">assume </span><a NAME="E2:136_1"/><span class="lab"><font color="Green" title="E140">A19</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <font color="Maroon" title="c5">g</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</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="c5">g</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</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#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</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> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</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="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span><br/><a NAME="E3:136_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E141">A20</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</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> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:136"><span class="lab"><font color="Green" title="E125">A4</font></span></a>, <a class="txt" href="#E7:136"><span class="lab"><font color="Green" title="E127">A6</font></span></a>, <a class="txt" href="#E11:136"><span class="lab"><font color="Green" title="E129">A8</font></span></a>, <a class="txt" href="#E20:136"><span class="lab"><font color="Green" title="E135">A14</font></span></a>, <a class="ref" href="mesfunc5.html#T22" target="_self" title="MESFUNC5:th.22">Th28</a></span>;<br/><span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:136"><span class="lab"><font color="Green" title="E125">A4</font></span></a>, <a class="txt" href="#E17:136"><span class="lab"><font color="Green" title="E133">A12</font></span></a>, <a class="txt" href="#E2:136_1"><span class="lab"><font color="Green" title="E140">A19</font></span></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:136"><span class="lab"><font color="Green" title="E125">A4</font></span></a>, <a class="txt" href="#E14:136"><span class="lab"><font color="Green" title="E131">A10</font></span></a>, <a class="txt" href="#E2:136_1"><span class="lab"><font color="Green" title="E140">A19</font></span></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:136"><span class="lab"><font color="Green" title="E125">A4</font></span></a>, <a class="txt" href="#E19:136"><span class="lab"><font color="Green" title="E134">A13</font></span></a>, <a class="txt" href="#E2:136_1"><span class="lab"><font color="Green" title="E140">A19</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:136"><span class="lab"><font color="Green" title="E125">A4</font></span></a>, <a class="txt" href="#E22:136"><span class="lab"><font color="Green" title="E137">A16</font></span></a>, <a class="txt" href="#E2:136_1"><span class="lab"><font color="Green" title="E140">A19</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</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> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</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="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:136_1"><span class="lab"><font color="Green" title="E141">A20</font></span></a>, <a class="ref" href="mesfunc1.html#D3" title="MESFUNC1:def.3">MESFUNC1:def 3</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:136_1"/>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</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="c5">g</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</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#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</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> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</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="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136_1"><span class="lab"><font color="Green" title="E140">A19</font></span></a>, <a class="txt" href="#E1:136_1"><span class="lab"><font color="Green" title="E139">A18</font></span></a>, <a class="ref" href="mesfunc1.html#D3" title="MESFUNC1:def.3">MESFUNC1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E24:136"/>
 ex <font color="Olive" title="b1">E1</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/>( <font color="Olive" title="b1">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136"><span class="lab"><font color="Green" title="E123">A2</font></span></a>, <a class="ref" href="mesfunc5.html#D17" target="_self" title="MESFUNC5:def.17">Def17</a></span>;<br/>
<a NAME="E25:136"/><span class="kw">then </span>
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136"><span class="lab"><font color="Green" title="E123">A2</font></span></a>, <a class="ref" href="mesfunc5.html#T100" target="_self" title="MESFUNC5:th.100">Th106</a></span>;<br/>
<a NAME="E26:136"/><span class="kw">then </span>
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#T97" target="_self" title="MESFUNC5:th.97">Th103</a></span>;<br/>
<a NAME="E27:136"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</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> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:136"><span class="lab"><font color="Green" title="E128">A7</font></span></a>, <a class="txt" href="#E7:136"><span class="lab"><font color="Green" title="E127">A6</font></span></a>, <a class="txt" href="#E11:136"><span class="lab"><font color="Green" title="E129">A8</font></span></a>, <a class="ref" href="mesfunc5.html#T106" target="_self" title="MESFUNC5:th.106">Th112</a></span>;<br/>
<span class="kw">hence </span><a NAME="E28:136"/>
<font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:136"><span class="lab"><font color="Green" title="E125">A4</font></span></a>, <a class="txt" href="#E6:136"><span class="lab"><font color="Green" title="E126">A5</font></span></a>, <a class="txt" href="#E23:136"><span class="lab"><font color="Green" title="E138">A17</font></span></a>, <a class="txt" href="#E21:136"><span class="lab"><font color="Green" title="E136">A15</font></span></a>, <a class="ref" href="mesfunc5.html#T102" target="_self" title="MESFUNC5:th.102">Th108</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
