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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#K4" title="MEASURE4:func.4">sigma_Meas</a> <font color="Olive" title="b1">C</font> <a href="measure3.html#R1" title="MEASURE3:pred.1">is_complete</a>  <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#K4" title="MEASURE4:func.4">sigma_Meas</a> <font color="Maroon" title="c2">C</font> <a href="measure3.html#R1" title="MEASURE3:pred.1">is_complete</a>  <a href="measure4.html#K1" title="MEASURE4:func.1">sigma_Field</a> <font color="Maroon" title="c2">C</font></span><br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="measure3.html#D2" title="MEASURE3:def.2">MEASURE3:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <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> or  not <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive">b<sub>1</sub></font> or  not <span class="p1">(<span class="default"><a href="measure4.html#K4" title="MEASURE4:func.4">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  or <font color="Maroon" title="c3">A</font> <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><br/><b>let </b><font color="Maroon" title="c4">B</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"> (  not <font color="Maroon" title="c4">B</font> <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> or  not <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font> or  not <span class="p1">(<span class="default"><a href="measure4.html#K4" title="MEASURE4:func.4">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  or <font color="Maroon" title="c3">A</font> <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><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:27"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c4">B</font> <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>and </b><br/><a NAME="E2:27"/><i><font color="Green" title="E25">A2</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font> &amp; <span class="p1">(<span class="default"><a href="measure4.html#K4" title="MEASURE4:func.4">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 ; <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="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><br/>

<b>reconsider </b><font color="Maroon" title="c5">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> <b>by </b><i><a class="txt" href="measure4.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a></i>;<br/>
<a NAME="E4:27"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c2">C</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">B</font> <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="measure4.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="measure4.html#E2:27"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="measure4.html#D4" target="_self" title="MEASURE4:def.4">Def4</a></i>;<br/>
<a NAME="E5:27"/>
<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/>
<a NAME="E6:27"/><b>then </b>
(  <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> <font color="Maroon" title="c2">C</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c2">C</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 
<b>by </b><i><a class="txt" href="measure4.html#E2:27"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="measure4.html#E4:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="measure4.html#D2" target="_self" title="MEASURE4:def.2">Def2</a>, <a class="ref" href="measure1.html#D4" title="MEASURE1:def.4">MEASURE1:def 4</a></i>;<br/>
<a NAME="E7:27"/><b>then </b>
<font color="Maroon" title="c2">C</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
;<br/>
<b>hence </b><a NAME="E8:27"/>
<font color="Maroon" title="c3">A</font> <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="measure4.html#T24" target="_self" title="MEASURE4:th.24">Th24</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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