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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><br/>  for <font color="Olive" title="b6">x</font>, <font color="Olive" title="b7">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K24" title="LPSPACC1:func.24">L-1-CSpace</a> <font color="Olive" title="b3">M</font></span>)</span> holds <br/> ( ( <font color="Olive" title="b4">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b6">x</font> &amp; <font color="Olive" title="b5">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b7">y</font> implies <font color="Olive" title="b4">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Olive" title="b5">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b6">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b7">y</font> ) &amp; ( <font color="Olive" title="b4">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b6">x</font> implies <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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b6">x</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><br/>  for <font color="Olive" title="b5">x</font>, <font color="Olive" title="b6">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K24" title="LPSPACC1:func.24">L-1-CSpace</a> <font color="Olive" title="b2">M</font></span>)</span> holds <br/> ( ( <font color="Olive" title="b3">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b5">x</font> &amp; <font color="Olive" title="b4">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b6">y</font> implies <font color="Olive" title="b3">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Olive" title="b4">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b5">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b6">y</font> ) &amp; ( <font color="Olive" title="b3">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b5">x</font> implies <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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b5">x</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><br/>  for <font color="Olive" title="b4">x</font>, <font color="Olive" title="b5">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K24" title="LPSPACC1:func.24">L-1-CSpace</a> <font color="Olive" title="b1">M</font></span>)</span> holds <br/> ( ( <font color="Olive" title="b2">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">x</font> &amp; <font color="Olive" title="b3">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b5">y</font> implies <font color="Olive" title="b2">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Olive" title="b3">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b5">y</font> ) &amp; ( <font color="Olive" title="b2">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">x</font> implies <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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b4">x</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><br/>  for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K24" title="LPSPACC1:func.24">L-1-CSpace</a> <font color="Maroon" title="c5">M</font></span>)</span> holds <br/> ( ( <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">x</font> &amp; <font color="Olive" title="b2">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">y</font> implies <font color="Olive" title="b1">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Olive" title="b2">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b4">y</font> ) &amp; ( <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">x</font> implies <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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b3">x</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">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K24" title="LPSPACC1:func.24">L-1-CSpace</a> <font color="Maroon" title="c5">M</font></span>)</span> holds <br/> ( ( <font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c7">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">y</font> implies <font color="Maroon" title="c6">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c7">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> ) &amp; ( <font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">x</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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K24" title="LPSPACC1:func.24">L-1-CSpace</a> <font color="Maroon" title="c5">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">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c7">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">y</font> implies <font color="Maroon" title="c6">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c7">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">y</font> ) &amp; ( <font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">x</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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c8">x</font> ) )</span><br/>















<span class="kw">set </span><font color="Maroon" title="c10">C</font> =  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c5">M</font>;<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="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">x</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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c8">x</font> )</span>
<div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c11">x1</font> = <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c12">y1</font> = <font color="Maroon" title="c9">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K22" title="LPSPACC1:func.22">Pre-L-CSpace</a> <font color="Maroon" title="c5">M</font></span>)</span> ;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:82_1"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">and </span><br/><a NAME="E3:82_1"/><span class="lab"><font color="Green" title="E52">A2</font></span>: 
<font color="Maroon" title="c7">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">y</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="c6">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c7">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">y</font></span><br/><a NAME="E4:82_1"/>
<font color="Maroon" title="c12">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K22" title="LPSPACC1:func.22">Pre-L-CSpace</a> <font color="Maroon" title="c5">M</font></span>)</span>
 ;<br/><a NAME="E5:82_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A3</font></span>: 
<font color="Maroon" title="c12">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c5">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#D19" target="_self" onmouseover="rs('lpspacc1/D19')" onmouseout="rh()">Def19</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">b</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><span class="kw"> such that </span><br/><a NAME="E7:82_1"/><span class="lab"><font color="Green" title="E54">A4</font></span>: 
<font color="Maroon" title="c12">y1</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="c13">b</font>,<font color="Maroon" title="c5">M</font>)
 <span class="kw">and </span><br/><a NAME="E8:82_1"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
<font color="Maroon" title="c13">b</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>
 ;<br/><a NAME="E9:82_1"/><span class="lab"><font color="Green" title="E56">A6</font></span>: 
<font color="Maroon" title="c13">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:82_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="txt" href="#E8:82_1"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>;<br/><a NAME="E10:82_1"/>
 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="c3">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> st <br/>( <font color="Maroon" title="c7">g</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="c5">M</font> &amp; <font color="Maroon" title="c13">b</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="c13">b</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="c5">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82_1"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E7:82_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a></span>;<br/><a NAME="E11:82_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A7</font></span>: 
 <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c13">b</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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T32" target="_self" onmouseover="rs('lpspacc1/T32')" onmouseout="rh()">Th32</a></span>;<br/><a NAME="E12:82_1"/>
<font color="Maroon" title="c11">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K22" title="LPSPACC1:func.22">Pre-L-CSpace</a> <font color="Maroon" title="c5">M</font></span>)</span>
 ;<br/><a NAME="E13:82_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A8</font></span>: 
<font color="Maroon" title="c11">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c5">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#D19" target="_self" onmouseover="rs('lpspacc1/D19')" onmouseout="rh()">Def19</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">a</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><span class="kw"> such that </span><br/><a NAME="E15:82_1"/><span class="lab"><font color="Green" title="E59">A9</font></span>: 
<font color="Maroon" title="c11">x1</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="c14">a</font>,<font color="Maroon" title="c5">M</font>)
 <span class="kw">and </span><br/><a NAME="E16:82_1"/><span class="lab"><font color="Green" title="E60">A10</font></span>: 
<font color="Maroon" title="c14">a</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>
 ;<br/><a NAME="E17:82_1"/>
<font color="Maroon" title="c14">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:82_1"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="txt" href="#E16:82_1"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>;<br/><a NAME="E18:82_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c5">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c12">y1</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="c14">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c13">b</font></span>)</span>,<font color="Maroon" title="c5">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:82_1"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="txt" href="#E5:82_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E9:82_1"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="ref" href="lpspacc1.html#D16" target="_self" onmouseover="rs('lpspacc1/D16')" onmouseout="rh()">Def16</a></span>;<br/><a NAME="E19:82_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A11</font></span>: 
<font color="Maroon" title="c11">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">y1</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="c14">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c13">b</font></span>)</span>,<font color="Maroon" title="c5">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#D19" target="_self" onmouseover="rs('lpspacc1/D19')" onmouseout="rh()">Def19</a></span>;<br/><a NAME="E20:82_1"/>
 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="c3">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> st <br/>( <font color="Maroon" title="c6">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="c5">M</font> &amp; <font color="Maroon" title="c14">a</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="c14">a</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="c5">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E15:82_1"><span class="lab"><font color="Green" title="E59">A9</font></span></a></span>;<br/><a NAME="E21:82_1"/><span class="kw">then </span>
 <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c14">a</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="c6">f</font>,<font color="Maroon" title="c5">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T32" target="_self" onmouseover="rs('lpspacc1/T32')" onmouseout="rh()">Th32</a></span>;<br/><a NAME="E22:82_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="c14">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c13">b</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="c6">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</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="#E2:82_1"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E3:82_1"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E15:82_1"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="txt" href="#E16:82_1"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="txt" href="#E7:82_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="txt" href="#E8:82_1"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="txt" href="#E11:82_1"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="ref" href="lpspacc1.html#T34" target="_self" onmouseover="rs('lpspacc1/T34')" onmouseout="rh()">Th34</a></span>;<br/><span class="kw">hence </span><a NAME="E23:82_1"/>
<font color="Maroon" title="c6">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c7">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E3:82_1"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E15:82_1"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="txt" href="#E7:82_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="txt" href="#E19:82_1"><span class="lab"><font color="Green" title="E61">A11</font></span></a>, <a class="ref" href="lpspacc1.html#T17" target="_self" onmouseover="rs('lpspacc1/T17')" onmouseout="rh()">Th17</a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</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>

<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"> verum</span>
<div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c11">x1</font> = <font color="Maroon" title="c8">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K22" title="LPSPACC1:func.22">Pre-L-CSpace</a> <font color="Maroon" title="c5">M</font></span>)</span> ;<br/><a NAME="E2:82_2"/>
<font color="Maroon" title="c11">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K22" title="LPSPACC1:func.22">Pre-L-CSpace</a> <font color="Maroon" title="c5">M</font></span>)</span>
 ;<br/><a NAME="E3:82_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A12</font></span>: 
<font color="Maroon" title="c11">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c5">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#D19" target="_self" onmouseover="rs('lpspacc1/D19')" onmouseout="rh()">Def19</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">f1</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><span class="kw"> such that </span><br/><a NAME="E5:82_2"/><span class="lab"><font color="Green" title="E52">A13</font></span>: 
<font color="Maroon" title="c11">x1</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="c12">f1</font>,<font color="Maroon" title="c5">M</font>)
 <span class="kw">and </span><br/><a NAME="E6:82_2"/><span class="lab"><font color="Green" title="E53">A14</font></span>: 
<font color="Maroon" title="c12">f1</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>
 ;<br/><a NAME="E7:82_2"/>
<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c5">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c11">x1</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="c12">f1</font></span>)</span>,<font color="Maroon" title="c5">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82_2"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="txt" href="#E6:82_2"><span class="lab"><font color="Green" title="E53">A14</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a>, <a class="txt" href="#E3:82_2"><span class="lab"><font color="Green" title="E51">A12</font></span></a>, <a class="ref" href="lpspacc1.html#D18" target="_self" onmouseover="rs('lpspacc1/D18')" onmouseout="rh()">Def18</a></span>;<br/><a NAME="E8:82_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A15</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x1</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="c12">f1</font></span>)</span>,<font color="Maroon" title="c5">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#D19" target="_self" onmouseover="rs('lpspacc1/D19')" onmouseout="rh()">Def19</a></span>;<br/><span class="kw">assume </span><a NAME="E9:82_2"/><span class="lab"><font color="Green" title="E55">A16</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">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"> <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> <font color="Maroon" title="c2">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c8">x</font></span><br/><a NAME="E10:82_2"/><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="c3">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> st <br/>( <font color="Maroon" title="c6">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="c5">M</font> &amp; <font color="Maroon" title="c12">f1</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="c12">f1</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="c5">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82_2"><span class="lab"><font color="Green" title="E52">A13</font></span></a></span>;<br/><a NAME="E11:82_2"/><span class="kw">then </span>
 <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c12">f1</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="c6">f</font>,<font color="Maroon" title="c5">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T32" target="_self" onmouseover="rs('lpspacc1/T32')" onmouseout="rh()">Th32</a></span>;<br/><a NAME="E12:82_2"/><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">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c12">f1</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="c6">f</font></span>)</span>,<font color="Maroon" title="c5">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:82_2"><span class="lab"><font color="Green" title="E55">A16</font></span></a>, <a class="txt" href="#E5:82_2"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="txt" href="#E6:82_2"><span class="lab"><font color="Green" title="E53">A14</font></span></a>, <a class="ref" href="lpspacc1.html#T35" target="_self" onmouseover="rs('lpspacc1/T35')" onmouseout="rh()">Th35</a></span>;<br/><span class="kw">hence </span><a NAME="E13:82_2"/>
<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> <font color="Maroon" title="c2">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:82_2"><span class="lab"><font color="Green" title="E55">A16</font></span></a>, <a class="txt" href="#E5:82_2"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="txt" href="#E8:82_2"><span class="lab"><font color="Green" title="E54">A15</font></span></a>, <a class="ref" href="lpspacc1.html#T18" target="_self" onmouseover="rs('lpspacc1/T18')" onmouseout="rh()">Th18</a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</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>


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