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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">g</font>, <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#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b5">k</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  ex <font color="Olive" title="b6">E</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/>( <font color="Olive" title="b2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b6">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> <font color="Olive" title="b3">g</font> &amp; <font color="Olive" title="b3">g</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b6">E</font> ) &amp;  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> (<font color="Olive" title="b4">f</font>,<font color="Olive" title="b2">M</font>,<font color="Olive" title="b5">k</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> (<font color="Olive" title="b4">f</font>,<font color="Olive" title="b2">M</font>,<font color="Olive" title="b5">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> (<font color="Olive" title="b3">g</font>,<font color="Olive" title="b2">M</font>,<font color="Olive" title="b5">k</font>) holds <br/><font color="Olive" title="b4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Olive" title="b3">g</font>,<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">g</font>, <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#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b4">k</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  ex <font color="Olive" title="b5">E</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">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b5">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> <font color="Olive" title="b2">g</font> &amp; <font color="Olive" title="b2">g</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b5">E</font> ) &amp;  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> (<font color="Olive" title="b3">f</font>,<font color="Olive" title="b1">M</font>,<font color="Olive" title="b4">k</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> (<font color="Olive" title="b3">f</font>,<font color="Olive" title="b1">M</font>,<font color="Olive" title="b4">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> (<font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">M</font>,<font color="Olive" title="b4">k</font>) holds <br/><font color="Olive" title="b3">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Olive" title="b2">g</font>,<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">g</font>, <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">k</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  ex <font color="Olive" title="b4">E</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="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b4">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> <font color="Olive" title="b1">g</font> &amp; <font color="Olive" title="b1">g</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b4">E</font> ) &amp;  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> (<font color="Olive" title="b2">f</font>,<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b3">k</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> (<font color="Olive" title="b2">f</font>,<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b3">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> (<font color="Olive" title="b1">g</font>,<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b3">k</font>) holds <br/><font color="Olive" title="b2">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Olive" title="b1">g</font>,<font color="Maroon" title="c3">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">g</font>, <font color="Maroon" title="c5">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">k</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  ex <font color="Olive" title="b2">E</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="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">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> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b2">E</font> ) &amp;  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font>) holds <br/><font color="Maroon" title="c5">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c3">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">k</font> be   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <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"> (  ex <font color="Olive" title="b1">E</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="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">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> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font> ) &amp;  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">k</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">k</font>) implies <font color="Maroon" title="c5">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c3">M</font> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:55"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
 ex <font color="Olive" title="b1">E</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="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">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> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font> )
 <span class="kw">and </span><br/><a NAME="E2:55"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
 <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">k</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E3:55"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
 <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">k</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="c5">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c3">M</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:55"/><span class="lab"><font color="Green" title="E46">A4</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">r</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><span class="kw"> such that </span><br/><a NAME="E7:55"/><span class="lab"><font color="Green" title="E47">A5</font></span>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> &amp; <font color="Maroon" title="c8">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">k</font>) &amp; <font color="Maroon" title="c5">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c3">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:55"><span class="lab"><font color="Green" title="E46">A4</font></span></a></span>;<br/>
<a NAME="E8:55"/>
<font color="Maroon" title="c8">r</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E3:55"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E5:55"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E7:55"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="lpspace2.html#T37" target="_self" title="LPSPACE2:th.37">Th37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:55"/>
<font color="Maroon" title="c5">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:55"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="lpspace1.html#T30" title="LPSPACE1:th.30">LPSPACE1:30</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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