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<span class="kw">set </span><font color="Maroon" title="c5">C</font> =  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">f1</font>, <font color="Maroon" title="c7">f2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>,<span class="p2">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</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"> ( (  for <font color="Olive" title="b1">z</font> being   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a><br/>  for <font color="Olive" title="b2">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font><br/>  for <font color="Olive" title="b3">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 <font color="Olive" title="b3">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">A</font> holds <br/><font color="Maroon" title="c6">f1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">A</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="Olive" title="b1">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b3">f</font></span>)</span>,<font color="Maroon" title="c4">M</font>) ) &amp; (  for <font color="Olive" title="b1">z</font> being   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a><br/>  for <font color="Olive" title="b2">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font><br/>  for <font color="Olive" title="b3">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 <font color="Olive" title="b3">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">A</font> holds <br/><font color="Maroon" title="c7">f2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">A</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="Olive" title="b1">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b3">f</font></span>)</span>,<font color="Maroon" title="c4">M</font>) ) implies <font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:69_1_2"/><span class="lab"><font color="Green" title="E39">A8</font></span>: 
 for <font color="Olive" title="b1">z</font> being   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a><br/>  for <font color="Olive" title="b2">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font><br/>  for <font color="Olive" title="b3">a</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">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">A</font> holds <br/><font color="Maroon" title="c6">f1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">A</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="Olive" title="b1">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b3">a</font></span>)</span>,<font color="Maroon" title="c4">M</font>)
 <span class="kw">and </span><br/><a NAME="E2:69_1_2"/><span class="lab"><font color="Green" title="E40">A9</font></span>: 
 for <font color="Olive" title="b1">z</font> being   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a><br/>  for <font color="Olive" title="b2">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font><br/>  for <font color="Olive" title="b3">a</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">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">A</font> holds <br/><font color="Maroon" title="c7">f2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">A</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="Olive" title="b1">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b3">a</font></span>)</span>,<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="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f2</font></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="69_1_2_1"><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">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <br/>  for <font color="Olive" title="b2">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font> holds  <font color="Maroon" title="c6">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">A</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font> holds  <font color="Maroon" title="c6">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c8">z</font>,<font color="Olive" title="b1">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c8">z</font>,<font color="Olive" title="b1">A</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c9">A</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</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"> <font color="Maroon" title="c6">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c9">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c9">A</font>)</span><br/><a NAME="E1:69_1_2_1"/>
<font color="Maroon" title="c9">A</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="c4">M</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">a1</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="E3:69_1_2_1"/><span class="lab"><font color="Green" title="E41">A10</font></span>: 
( <font color="Maroon" title="c9">A</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="c10">a1</font>,<font color="Maroon" title="c4">M</font>) &amp; <font color="Maroon" title="c10">a1</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">thus </span><font color="Maroon" title="c6">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c9">A</font>) = 
 <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c10">a1</font></span>)</span>,<font color="Maroon" title="c4">M</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E3:69_1_2_1"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>
<br/>.= 
<font color="Maroon" title="c7">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c9">A</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69_1_2"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="txt" href="#E3:69_1_2_1"><span class="lab"><font color="Green" title="E41">A10</font></span></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>
<span class="kw">hence </span><a NAME="E4:69_1_2"/>
<font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f2</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/>


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