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<span class="kw">let </span><font color="Maroon" title="c2">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="c2">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="c2">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> holds <br/>(  <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>) iff <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> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">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="c3">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="c2">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> holds <br/>(  <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>) iff <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> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">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="c2">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="c4">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="c4">M</font> holds <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>  <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="c4">M</font>) iff <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="c4">M</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font> be   <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>; <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="c5">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> &amp; <font color="Maroon" title="c6">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="c4">M</font> implies (  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">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="c6">g</font>,<font color="Maroon" title="c4">M</font>) iff <font color="Maroon" title="c5">f</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">M</font> ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:62"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<font color="Maroon" title="c5">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 class="kw">and </span><br/><a NAME="E2:62"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<font color="Maroon" title="c6">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="c4">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> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">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="c6">g</font>,<font color="Maroon" title="c4">M</font>) iff <font color="Maroon" title="c5">f</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">M</font> )</span><br/>

<div><span class="kw">hereby </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="c5">f</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">M</font> implies  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">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="c6">g</font>,<font color="Maroon" title="c4">M</font>) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:62_1"/>
 <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">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="c6">g</font>,<font color="Maroon" title="c4">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="c5">f</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">M</font></span><br/><a NAME="E2:62_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</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">r</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> &amp; <font color="Maroon" title="c6">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="c4">M</font> &amp; <font color="Maroon" title="c6">g</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Olive" title="b1">r</font>,<font color="Maroon" title="c4">M</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>;<br/><a NAME="E3:62_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">r</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/>( <font color="Maroon" title="c5">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</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> &amp; <font color="Maroon" title="c6">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="c4">M</font> &amp; <font color="Maroon" title="c6">g</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Olive" title="b1">r</font>,<font color="Maroon" title="c4">M</font> )
 ;<br/><span class="kw">hence </span><a NAME="E4:62_1"/>
<font color="Maroon" title="c5">f</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">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"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E4:62"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">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> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">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="c6">g</font>,<font color="Maroon" title="c4">M</font>)</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_2"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">M</font>) holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">M</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">M</font>) implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">M</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:62_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">M</font>)</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">r</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><span class="kw"> such that </span><br/><a NAME="E3:62_2"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> &amp; <font color="Maroon" title="c8">r</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 class="kw">and </span><br/><a NAME="E4:62_2"/>
<font color="Maroon" title="c5">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 class="kw">and </span><br/><a NAME="E5:62_2"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c4">M</font>
 ;<br/><a NAME="E6:62_2"/>
<font color="Maroon" title="c6">g</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62"><span class="lab"><font color="Green" title="E38">A3</font></span></a></span>;<br/><a NAME="E7:62_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">g</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:62_2"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="ref" href="lpspacc1.html#T24" target="_self" onmouseover="rs('lpspacc1/T24')" onmouseout="rh()">Th24</a></span>;<br/><span class="kw">hence </span><a NAME="E8:62_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E3:62_2"><span class="lab"><font color="Green" title="E39">A4</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/></div><span class="kw">end;</span></div>
<a NAME="E6:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A6</font></span>: 
 <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">M</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">M</font>)
 
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_3"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">M</font>) holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">M</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">M</font>) implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">M</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:62_3"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">M</font>)</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">r</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><span class="kw"> such that </span><br/><a NAME="E3:62_3"/><span class="lab"><font color="Green" title="E40">A7</font></span>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> &amp; <font color="Maroon" title="c8">r</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 class="kw">and </span><br/><a NAME="E4:62_3"/>
<font color="Maroon" title="c6">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="c4">M</font>
 <span class="kw">and </span><br/><a NAME="E5:62_3"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
<font color="Maroon" title="c6">g</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c4">M</font>
 ;<br/><a NAME="E6:62_3"/>
<font color="Maroon" title="c5">f</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E5:62_3"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="lpspacc1.html#T24" target="_self" onmouseover="rs('lpspacc1/T24')" onmouseout="rh()">Th24</a></span>;<br/><span class="kw">hence </span><a NAME="E7:62_3"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E3:62_3"><span class="lab"><font color="Green" title="E40">A7</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/></div><span class="kw">end;</span></div>
<a NAME="E8:62"/><span class="kw">then </span>
 <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">M</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">M</font>)
 
;<br/>
<span class="kw">hence </span><a NAME="E9:62"/>
 <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">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="c6">g</font>,<font color="Maroon" title="c4">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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/>


</div>
