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<span class="kw">set </span><font color="Maroon" title="c5">C</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><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="c4">M</font>)</span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> : <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="c4">M</font> </span> } </span> ;<br/>
<a NAME="E1:66_1_1"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><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="c4">M</font>)</span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> : <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="c4">M</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="66_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><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="c4">M</font>)</span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> : <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="c4">M</font> </span> } </span>  or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:66_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><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="c4">M</font>)</span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> : <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="c4">M</font> </span> } </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"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font></span>)</span></span><br/>

<a NAME="E2:66_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> st <br/>(  <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="c4">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; <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="c4">M</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:66_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font></span>)</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/>


</div><span class="kw">end;</span></div>
<a NAME="E2:66_1_1"/>
<font color="Maroon" title="c2">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <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="c4">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="lpspacc1.html#E29"><span class="lab"><font color="Green" title="E18">Lm3</font></span></a></span>;<br/>
<a NAME="E3:66_1_1"/><span class="kw">then </span>
 <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">X</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a></span>)</span>,<font color="Maroon" title="c4">M</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><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="c4">M</font>)</span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> : <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="c4">M</font> </span> } </span> 
 
;<br/>
<span class="kw">hence </span><a NAME="E4:66_1_1"/>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><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="c4">M</font>)</span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> : <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="c4">M</font> </span> } </span>  is   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_1"><span class="lab"><font color="Green" title="E38">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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