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<span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">S</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Olive" title="b1">X</font><br/>  for <font color="Olive" title="b3">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b2">S</font><br/>  for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Olive" title="b1">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Olive" title="b4">f</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Olive" title="b5">g</font>,<font color="Olive" title="b3">M</font> holds <br/><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b4">f</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c2">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b5">g</font>,<font color="Olive" title="b3">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c3">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">f</font>, <font color="Olive" title="b4">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c3">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Olive" title="b3">f</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Olive" title="b4">g</font>,<font color="Olive" title="b2">M</font> holds <br/><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b3">f</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c2">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b4">g</font>,<font color="Olive" title="b2">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c3">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c4">S</font><br/>  for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c3">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Olive" title="b2">f</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Olive" title="b3">g</font>,<font color="Olive" title="b1">M</font> holds <br/><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b2">f</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c2">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b3">g</font>,<font color="Olive" title="b1">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c4">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c3">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Olive" title="b1">f</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Olive" title="b2">g</font>,<font color="Maroon" title="c5">M</font> holds <br/><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b1">f</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c2">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b2">g</font>,<font color="Maroon" title="c5">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">f</font>, <font color="Maroon" title="c7">g</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c3">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">f</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c5">M</font> implies <font color="Maroon" title="c2">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c6">f</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c2">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c5">M</font> )</span><br/>











<span class="kw">assume </span><a NAME="E1:50"/>
<font color="Maroon" title="c6">f</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c5">M</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c6">f</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c2">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c5">M</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S</font><span class="kw"> such that </span><br/><a NAME="E3:50"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c5">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E4:50"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 ;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> = 
<font color="Maroon" title="c2">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="rfunct_1.html#T49" onmouseover="rs('rfunct_1/T49')" onmouseout="rh()">RFUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T49" onmouseover="rs('rfunct_1/T49')" onmouseout="rh()">RFUNCT_1:49</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:50"/>
<font color="Maroon" title="c2">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c6">f</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c2">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c5">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E31">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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