<?xml version="1.0"?>
<div class="add">

<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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> <br/> for <font color="Olive" title="b3">A</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font> holds <br/> ( <font color="Olive" title="b2">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b3">A</font> iff for <font color="Olive" title="b4">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Olive" title="b3">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b4">r</font></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Olive" title="b1">S</font> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> <br/> for <font color="Olive" title="b2">A</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b2">A</font> iff for <font color="Olive" title="b3">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Olive" title="b2">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">r</font></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font> )</span><br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> holds <br/> ( <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b1">A</font> iff for <font color="Olive" title="b2">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Olive" title="b1">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Maroon" title="c3">f</font>,<font color="Olive" title="b2">r</font></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font> )</span><br/><b>let </b><font color="Maroon" title="c4">A</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font> iff for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Maroon" title="c3">f</font>,<font color="Olive" title="b1">r</font></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font> )</span><br/>







<a NAME="E1:15"/><i><font color="Green" title="E6">A1</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font> iff  <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font> )
 
<b>by </b><i><a class="ref" href="mesfunc6.html#D6" target="_self" title="MESFUNC6:def.6">Def6</a></i>;<br/>
<b>thus </b><a NAME="E2:15"/>
( <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font> implies for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Maroon" title="c3">f</font>,<font color="Olive" title="b1">r</font></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Maroon" title="c3">f</font>,<font color="Olive" title="b1">r</font></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font> ) implies <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Maroon" title="c3">f</font>,<font color="Olive" title="b1">r</font></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font></span><br/>

<b>let </b><font color="Maroon" title="c5">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">r</font></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font></span><br/>

<a NAME="E2:15_1"/>
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font>
 
;<br/>
<b>hence </b><a NAME="E3:15_1"/>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">r</font></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc6.html#E1:15"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="mesfunc6.html#E1:15_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="mesfunc1.html#T32" title="MESFUNC1:th.32">MESFUNC1:32</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:15"/>
for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Maroon" title="c3">f</font>,<font color="Olive" title="b1">r</font></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font></span><br/>

<a NAME="E4:15"/><b>then </b>
for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_eq_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Olive" title="b1">r</font></span>)</span></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font>
 
;<br/>
<b>hence </b><a NAME="E5:15"/>
<font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="mesfunc6.html#E1:15"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="mesfunc1.html#T32" title="MESFUNC1:th.32">MESFUNC1:32</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
