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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="Red">::  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> holds   <a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Olive" title="b2">M</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lpspace2.html#K9" title="LPSPACE2:func.9">lmultCoset</a> <font color="Olive" title="b2">M</font>,1</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="Red">::  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> holds   <a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Olive" title="b1">M</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lpspace2.html#K9" title="LPSPACE2:func.9">lmultCoset</a> <font color="Olive" title="b1">M</font>,1</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="Red">::  thesis: </font></span></a><span class="hide">  <a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lpspace2.html#K9" title="LPSPACE2:func.9">lmultCoset</a> <font color="Maroon" title="c3">M</font>,1</span><br/>





<a NAME="E1:109"/><span class="lab"><font color="Green" title="E86">A1</font></span>: 
 <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T72" target="_self" title="LPSPACE2:th.72">Th9a</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font> holds  <span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">z</font>,<font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lpspace2.html#K9" title="LPSPACE2:func.9">lmultCoset</a> <font color="Maroon" title="c3">M</font>,1</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">z</font>,<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#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lpspace2.html#K9" title="LPSPACE2:func.9">lmultCoset</a> <font color="Maroon" title="c3">M</font>,1</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">A</font></span><br/><a NAME="E1:109_1"/>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c3">M</font></span>)</span> where <font color="Olive" title="b1">f</font> is   <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>  : <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c3">M</font> </span>}</span> 
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">a</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="E3:109_1"/><span class="lab"><font color="Green" title="E87">A2</font></span>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c3">M</font> )
 ;<br/><a NAME="E4:109_1"/><span class="lab"><font color="Green" title="E88">A3</font></span>: 
<font color="Maroon" title="c5">A</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T72" target="_self" title="LPSPACE2:th.72">Th9a</a></span>;<br/><a NAME="E5:109_1"/><span class="lab"><font color="Green" title="E89">A4</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1"><span class="lab"><font color="Green" title="E87">A2</font></span></a>, <a class="ref" href="lpspace1.html#T38" title="LPSPACE1:th.38">LPSPACE1:38</a></span>;<br/><a NAME="E6:109_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A5</font></span>: 
<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#D17" title="LPSPACE1:def.17">LPSPACE1:def 17</a></span>;<br/><a NAME="E7:109_1"/>
<font color="Maroon" title="c4">z</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1"><span class="lab"><font color="Green" title="E87">A2</font></span></a>, <a class="ref" href="lpspace1.html#T24" title="LPSPACE1:th.24">LPSPACE1:24</a></span>;<br/><a NAME="E8:109_1"/><span class="kw">then </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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">a</font></span>)</span> &amp; <font color="Maroon" title="c4">z</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">a</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="lpspace2.html#E92"><span class="lab"><font color="Green" title="E77">Lem00</font></span></a></span>;<br/><a NAME="E9:109_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">A</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c3">M</font>,1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:109_1"><span class="lab"><font color="Green" title="E90">A5</font></span></a>, <a class="txt" href="lpspace2.html#E96"><span class="lab"><font color="Green" title="E81">Lem03</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:109_1"/>
<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lpspace2.html#K9" title="LPSPACE2:func.9">lmultCoset</a> <font color="Maroon" title="c3">M</font>,1</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:109_1"><span class="lab"><font color="Green" title="E88">A3</font></span></a>, <a class="txt" href="#E5:109_1"><span class="lab"><font color="Green" title="E89">A4</font></span></a>, <a class="ref" href="lpspace2.html#D10" target="_self" title="LPSPACE2:def.10">VSPDef5X</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:109"/>
 <a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lpspace2.html#K9" title="LPSPACE2:func.9">lmultCoset</a> <font color="Maroon" title="c3">M</font>,1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:109"><span class="lab"><font color="Green" title="E86">A1</font></span></a>, <a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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