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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">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><br/>  for <font color="Olive" title="b4">A</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b2">f</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  &amp; <font color="Olive" title="b3">g</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  &amp; <font color="Olive" title="b2">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b4">A</font> &amp; <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">A</font> &amp; <font color="Olive" title="b4">A</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> <font color="Olive" title="b3">g</font> holds <br/><font color="Olive" title="b2">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</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">A</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">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><br/>  for <font color="Olive" title="b3">A</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">f</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  &amp; <font color="Olive" title="b1">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b3">A</font> &amp; <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">A</font> &amp; <font color="Olive" title="b3">A</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> <font color="Olive" title="b2">g</font> holds <br/><font color="Olive" title="b1">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</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">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">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">  for <font color="Olive" title="b1">A</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c3">f</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  &amp; <font color="Maroon" title="c4">g</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b1">A</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> <font color="Maroon" title="c4">g</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">A</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> ( <font color="Maroon" title="c3">f</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  &amp; <font color="Maroon" title="c4">g</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c5">A</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> <font color="Maroon" title="c4">g</font> implies <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:12"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> 
 <span class="kw">and </span><br/><a NAME="E2:12"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<font color="Maroon" title="c4">g</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> 
 <span class="kw">and </span><br/><a NAME="E3:12"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font>
 <span class="kw">and </span><br/><a NAME="E4:12"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
( <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c5">A</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> <font color="Maroon" title="c4">g</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="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font></span><br/>

<a NAME="E5:12"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c4">g</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="mesfunc2.html#T10" target="_self" title="MESFUNC2:th.10">Th12</a></span>;<br/>
<a NAME="E6:12"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="mesfunc1.html#T37" title="MESFUNC1:th.37">MESFUNC1:37</a></span>;<br/>
<a NAME="E7:12"/><span class="lab"><font color="Green" title="E16">A7</font></span>: 
 <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <font color="Maroon" title="c4">g</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:12"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="mesfunc2.html#T9" target="_self" title="MESFUNC2:th.9">Th11</a></span>;<br/>
<a NAME="E8:12"/>
 <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="mesfunc2.html#T9" target="_self" title="MESFUNC2:th.9">Th11</a></span>;<br/>
<a NAME="E9:12"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E7:12"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="ref" href="mesfunc2.html#T7" target="_self" title="MESFUNC2:th.7">Th7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:12"/>
<font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc2.html#T8" target="_self" title="MESFUNC2:th.8">Th9</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/>


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