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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#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">E1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b4">E2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">M</font>)<br/>  for <font color="Olive" title="b5">f</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#K6" title="NUMBERS:func.6">ExtREAL</a>  st <font color="Olive" title="b3">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">E2</font> &amp; <font color="Olive" title="b5">f</font> is <font color="Olive" title="b3">E1</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  holds <br/><font color="Olive" title="b5">f</font> is <font color="Olive" title="b4">E2</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">E1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b3">E2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">M</font>)<br/>  for <font color="Olive" title="b4">f</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#K6" title="NUMBERS:func.6">ExtREAL</a>  st <font color="Olive" title="b2">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">E2</font> &amp; <font color="Olive" title="b4">f</font> is <font color="Olive" title="b2">E1</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  holds <br/><font color="Olive" title="b4">f</font> is <font color="Olive" title="b3">E2</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <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><br/>  for <font color="Olive" title="b2">E2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>)<br/>  for <font color="Olive" title="b3">f</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#K6" title="NUMBERS:func.6">ExtREAL</a>  st <font color="Olive" title="b1">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">E2</font> &amp; <font color="Olive" title="b3">f</font> is <font color="Olive" title="b1">E1</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  holds <br/><font color="Olive" title="b3">f</font> is <font color="Olive" title="b2">E2</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">E1</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">E2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>)<br/>  for <font color="Olive" title="b2">f</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#K6" title="NUMBERS:func.6">ExtREAL</a>  st <font color="Maroon" title="c4">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">E2</font> &amp; <font color="Olive" title="b2">f</font> is <font color="Maroon" title="c4">E1</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  holds <br/><font color="Olive" title="b2">f</font> is <font color="Olive" title="b1">E2</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">E2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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> being   <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>  st <font color="Maroon" title="c4">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E2</font> &amp; <font color="Olive" title="b1">f</font> is <font color="Maroon" title="c4">E1</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  holds <br/><font color="Olive" title="b1">f</font> is <font color="Maroon" title="c5">E2</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c6">f</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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E2</font> &amp; <font color="Maroon" title="c6">f</font> is <font color="Maroon" title="c4">E1</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  implies <font color="Maroon" title="c6">f</font> is <font color="Maroon" title="c5">E2</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<font color="Maroon" title="c4">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E2</font>
 <span class="kw">and </span><br/><a NAME="E2:37"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<font color="Maroon" title="c6">f</font> is <font color="Maroon" title="c4">E1</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">f</font> is <font color="Maroon" title="c5">E2</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/>

<a NAME="E3:37"/>
 for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Maroon" title="c5">E2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K14" title="MESFUNC1:func.14">less_dom</a> (<font color="Maroon" title="c6">f</font>,<font color="Olive" title="b1">r</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">E2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K14" title="MESFUNC1:func.14">less_dom</a> (<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">r</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>)</span><br/>

<a NAME="E1:37_1"/>
<font color="Maroon" title="c4">E1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K14" title="MESFUNC1:func.14">less_dom</a> (<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">r</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="mesfunc1.html#D16" onmouseover="rs('mesfunc1/D16')" onmouseout="rh()">MESFUNC1:def 16</a></span>;<br/>
<a NAME="E2:37_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">E2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K14" title="MESFUNC1:func.14">less_dom</a> (<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">r</font>)</span>)</span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="mesfun14.html#T27" target="_self" onmouseover="rs('mesfun14/T27')" onmouseout="rh()">Th27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:37_1"/>
<font color="Maroon" title="c5">E2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K14" title="MESFUNC1:func.14">less_dom</a> (<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">r</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<span class="kw">hence </span><a NAME="E4:37"/>
<font color="Maroon" title="c6">f</font> is <font color="Maroon" title="c5">E2</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D16" onmouseover="rs('mesfunc1/D16')" onmouseout="rh()">MESFUNC1:def 16</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
