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<b>let </b><font color="Maroon" title="c1">X</font> be    <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">C</font> being   <a href="measure4.html#M1" title="MEASURE4:mode.1">C_Measure</a> of <font color="Maroon" title="c1">X</font> holds   <a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <font color="Olive" title="b1">C</font> is   <a href="measure1.html#NM1" title="MEASURE1:NM.1">Measure</a> of  <a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <font color="Olive" title="b1">C</font></span><br/><b>let </b><font color="Maroon" title="c2">C</font> be    <a href="measure4.html#M1" title="MEASURE4:mode.1">C_Measure</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">  <a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <font color="Maroon" title="c2">C</font> is   <a href="measure1.html#NM1" title="MEASURE1:NM.1">Measure</a> of  <a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <font color="Maroon" title="c2">C</font></span><br/>



<div><i><font color="Green" title="E21">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">A</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <font color="Maroon" title="c2">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">A</font></span><br/><b>reconsider </b><font color="Maroon" title="c4">A'</font> = <font color="Maroon" title="c3">A</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E2:22_1"/><i><font color="Green" title="E22">A2</font></i>: 
<span class="p1">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">A'</font>
 <b>by </b><i><a class="ref" href="measure4.html#D4" target="_self" title="MEASURE4:def.4">Def4</a></i>;<br/><a NAME="E3:22_1"/>
<font color="Maroon" title="c2">C</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="ref" href="measure4.html#D2" target="_self" title="MEASURE4:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E4:22_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="#E2:22_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="measure1.html#D4" title="MEASURE1:def.4">MEASURE1:def 4</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>
<a NAME="E2:22"/><i><font color="Green" title="E22">X</font></i>: 
<font color="Maroon" title="c2">C</font> is <a href="valued_0.html#V12" title="VALUED_0:attr.12">zeroed</a>
 
<b>by </b><i><a class="ref" href="measure4.html#D2" target="_self" title="MEASURE4:def.2">Def2</a></i>;<br/>
<a NAME="E3:22"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T10" title="PROB_1:th.10">PROB_1:10</a></i>;<br/>
<a NAME="E4:22"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="measure4.html#D4" target="_self" title="MEASURE4:def.4">Def4</a></i>;<br/>
<a NAME="E5:22"/><b>then </b><i><font color="Green" title="E23">A3</font></i>: 
(  <a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <font color="Maroon" title="c2">C</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <span class="p1">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 
<b>by </b><i><a class="txt" href="#E1:22"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="#E2:22"><i><font color="Green" title="E22">X</font></i></a>, <a class="ref" href="measure1.html#D4" title="MEASURE1:def.4">MEASURE1:def 4</a>, <a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <font color="Maroon" title="c2">C</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">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">B</font> implies <span class="p1">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">B</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:22_2"/><i><font color="Green" title="E24">A4</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">B</font></span>)</span></span><br/><b>reconsider </b><font color="Maroon" title="c5">A'</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c6">B'</font> = <font color="Maroon" title="c4">B</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E3:22_2"/><i><font color="Green" title="E25">A5</font></i>: 
( <span class="p1">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">A'</font> &amp; <span class="p1">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">B'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">B'</font> )
 <b>by </b><i><a class="ref" href="measure4.html#D4" target="_self" title="MEASURE4:def.4">Def4</a></i>;<br/><a NAME="E4:22_2"/>
<font color="Maroon" title="c2">C</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">B'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">A'</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">B'</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:22_2"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="measure4.html#T15" target="_self" title="MEASURE4:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E5:22_2"/>
<span class="p1">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:22_2"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="measure4.html#D4" target="_self" title="MEASURE4:def.4">Def4</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>hence </b><a NAME="E7:22"/>
 <a href="measure4.html#K3" title="MEASURE4:func.3">sigma_Meas</a> <font color="Maroon" title="c2">C</font> is   <a href="measure1.html#NM1" title="MEASURE1:NM.1">Measure</a> of  <a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <font color="Maroon" title="c2">C</font>
 <b>by </b><i><a class="txt" href="#E5:22"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="measure1.html#D5" title="MEASURE1:def.5">MEASURE1:def 5</a>, <a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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