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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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 <span class="p1">(<span class="default"><a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <font color="Olive" title="b1">C</font></span>)</span></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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 <span class="p1">(<span class="default"><a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <font color="Maroon" title="c2">C</font></span>)</span></span><br/>



<div><span class="lab"><font color="Green" title="E17">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">reconsider </span><font color="Maroon" title="c4">A9</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="E2:27_1"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c2">C</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure4.html#D2" target="_self" title="MEASURE4:def.2">Def2</a></span>;<br/><a NAME="E3:27_1"/>
<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">A9</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">A9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure4.html#D4" target="_self" title="MEASURE4:def.4">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E4:27_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="measure1.html#D4" title="MEASURE1:def.4">MEASURE1:def 4</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>
<a NAME="E2:27"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T10" title="PROB_1:th.10">PROB_1:10</a></span>;<br/>
<a NAME="E3:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure4.html#D4" target="_self" title="MEASURE4:def.4">Def4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E19">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">reconsider </span><font color="Maroon" title="c5">A9</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c6">B9</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="E2:27_2"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<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">B9</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">B9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure4.html#D4" target="_self" title="MEASURE4:def.4">Def4</a></span>;<br/><span class="kw">assume </span><a NAME="E3:27_2"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><a NAME="E4:27_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<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">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">B9</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">A9</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">B9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure4.html#T15" target="_self" title="MEASURE4:th.15">Th15</a></span>;<br/><a NAME="E5:27_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> <font color="Maroon" title="c5">A9</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">A9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure4.html#D4" target="_self" title="MEASURE4:def.4">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E6:27_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_2"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E4:27_2"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="measure4.html#D4" target="_self" title="MEASURE4:def.4">Def4</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>
<a NAME="E5:27"/>
<font color="Maroon" title="c2">C</font> is  <a href="valued_0.html#V12" title="VALUED_0:attr.12">zeroed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure4.html#D2" target="_self" title="MEASURE4:def.2">Def2</a></span>;<br/>
<a NAME="E6:27"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:27"/>
 <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 <span class="p1">(<span class="default"><a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <font color="Maroon" title="c2">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E4:27"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="measure1.html#D4" title="MEASURE1:def.4">MEASURE1:def 4</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></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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