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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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Olive" title="b3">M</font> &amp; <font color="Olive" title="b5">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Olive" title="b3">M</font> &amp;  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Olive" title="b4">f</font>,<font color="Olive" title="b3">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Olive" title="b5">g</font>,<font color="Olive" title="b3">M</font>) holds <br/> <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<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="Olive" title="b4">f</font></span>)</span>,<font color="Olive" title="b3">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<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="Olive" title="b5">g</font></span>)</span>,<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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Olive" title="b2">M</font> &amp; <font color="Olive" title="b4">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Olive" title="b2">M</font> &amp;  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Olive" title="b3">f</font>,<font color="Olive" title="b2">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Olive" title="b4">g</font>,<font color="Olive" title="b2">M</font>) holds <br/> <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<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="Olive" title="b3">f</font></span>)</span>,<font color="Olive" title="b2">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<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="Olive" title="b4">g</font></span>)</span>,<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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b3">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Olive" title="b1">M</font> &amp;  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Olive" title="b2">f</font>,<font color="Olive" title="b1">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Olive" title="b3">g</font>,<font color="Olive" title="b1">M</font>) holds <br/> <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<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="Olive" title="b2">f</font></span>)</span>,<font color="Olive" title="b1">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<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="Olive" title="b3">g</font></span>)</span>,<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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c5">M</font> &amp; <font color="Olive" title="b2">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c5">M</font> &amp;  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c5">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c5">M</font>) holds <br/> <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<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="Olive" title="b1">f</font></span>)</span>,<font color="Maroon" title="c5">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<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="Olive" title="b2">g</font></span>)</span>,<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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c5">M</font> &amp; <font color="Maroon" title="c7">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c5">M</font> &amp;  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c5">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c5">M</font>) implies  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<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>,<font color="Maroon" title="c5">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<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>,<font color="Maroon" title="c5">M</font>) )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:65"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
( <font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c5">M</font> &amp; <font color="Maroon" title="c7">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c5">M</font> )
 <span class="kw">and </span><br/><a NAME="E2:65"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
 <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c5">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</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">  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<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>,<font color="Maroon" title="c5">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<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>,<font color="Maroon" title="c5">M</font>)</span><br/>

<a NAME="E3:65"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="lpspacc1.html#T32" target="_self" onmouseover="rs('lpspacc1/T32')" onmouseout="rh()">Th32</a></span>;<br/>
<a NAME="E4:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A3</font></span>: 
<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="ref" href="lpspacc1.html#T26" target="_self" onmouseover="rs('lpspacc1/T26')" onmouseout="rh()">Th26</a></span>;<br/>
<a NAME="E5:65"/>
( <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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c5">M</font> &amp; <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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c5">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="lpspacc1.html#T18" target="_self" onmouseover="rs('lpspacc1/T18')" onmouseout="rh()">Th18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:65"/>
 <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<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>,<font color="Maroon" title="c5">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<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>,<font color="Maroon" title="c5">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="ref" href="lpspacc1.html#T32" target="_self" onmouseover="rs('lpspacc1/T32')" onmouseout="rh()">Th32</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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