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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> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">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><br/>  for <font color="Olive" title="b4">r</font>, <font color="Olive" title="b5">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <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> &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">f</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K14" title="MESFUNC1:func.14">great_dom</a> (<font color="Olive" title="b2">f</font>,<font color="Olive" title="b4">r</font>)</span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<font color="Olive" title="b2">f</font>,<font color="Olive" title="b5">s</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S</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> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">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><br/>  for <font color="Olive" title="b3">r</font>, <font color="Olive" title="b4">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <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> &amp; <font color="Olive" title="b2">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="b1">f</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K14" title="MESFUNC1:func.14">great_dom</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">r</font>)</span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b4">s</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font></span><br/><span class="kw">let </span><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="numbers.html#K1" title="NUMBERS:func.1">REAL</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#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">r</font>, <font color="Olive" title="b3">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <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> &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="c3">f</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K14" title="MESFUNC1:func.14">great_dom</a> (<font color="Maroon" title="c3">f</font>,<font color="Olive" title="b2">r</font>)</span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<font color="Maroon" title="c3">f</font>,<font color="Olive" title="b3">s</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <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> &amp; <font color="Maroon" title="c4">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="c3">f</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K14" title="MESFUNC1:func.14">great_dom</a> (<font color="Maroon" title="c3">f</font>,<font color="Olive" title="b1">r</font>)</span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<font color="Maroon" title="c3">f</font>,<font color="Olive" title="b2">s</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">r</font>, <font color="Maroon" title="c6">s</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <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="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c4">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="c3">f</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K14" title="MESFUNC1:func.14">great_dom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">r</font>)</span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">s</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> )</span><br/>











<a NAME="E1:19"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
(  <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> &amp;  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> )
 
;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:19"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<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 class="kw">and </span><br/><a NAME="E3:19"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
<font color="Maroon" title="c4">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="c3">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K14" title="MESFUNC1:func.14">great_dom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">r</font>)</span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">s</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font></span><br/>

<a NAME="E4:19"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="mesfunc6.html#D1" target="_self" title="MESFUNC6:def.1">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:19"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K14" title="MESFUNC1:func.14">great_dom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">r</font>)</span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">s</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="mesfunc1.html#T32" title="MESFUNC1:th.32">MESFUNC1:32</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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