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<b>let </b><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">X1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Y</font>, <font color="Olive" title="b2">Y1</font> being <a href="valued_2.html#V1" title="VALUED_2:attr.1">complex-functions-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <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>,<font color="Olive" title="b1">Y</font><br/> for <font color="Olive" title="b4">f1</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c3">X1</font>,<font color="Olive" title="b2">Y1</font><br/> for <font color="Olive" title="b5">g</font> being <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> <a href="valued_2.html#K24" title="VALUED_2:func.24">&lt;#&gt;</a> <font color="Olive" title="b5">g</font> holds <br/><font color="Olive" title="b3">f</font> <a href="valued_2.html#K24" title="VALUED_2:func.24">&lt;#&gt;</a> <span class="p1">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Olive" title="b5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_2.html#K14" title="VALUED_2:func.14">&lt;-&gt;</a> <font color="Olive" title="b4">f1</font></span><br/><b>let </b><font color="Maroon" title="c4">Y</font>, <font color="Maroon" title="c5">Y1</font> be  <a href="valued_2.html#V1" title="VALUED_2:attr.1">complex-functions-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <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>,<font color="Maroon" title="c4">Y</font><br/> for <font color="Olive" title="b2">f1</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c5">Y1</font><br/> for <font color="Olive" title="b3">g</font> being <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="valued_2.html#K24" title="VALUED_2:func.24">&lt;#&gt;</a> <font color="Olive" title="b3">g</font> holds <br/><font color="Olive" title="b1">f</font> <a href="valued_2.html#K24" title="VALUED_2:func.24">&lt;#&gt;</a> <span class="p1">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Olive" title="b3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_2.html#K14" title="VALUED_2:func.14">&lt;-&gt;</a> <font color="Olive" title="b2">f1</font></span><br/><b>let </b><font color="Maroon" title="c6">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c4">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f1</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c5">Y1</font><br/> for <font color="Olive" title="b2">g</font> being <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="valued_2.html#K24" title="VALUED_2:func.24">&lt;#&gt;</a> <font color="Olive" title="b2">g</font> holds <br/><font color="Maroon" title="c6">f</font> <a href="valued_2.html#K24" title="VALUED_2:func.24">&lt;#&gt;</a> <span class="p1">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_2.html#K14" title="VALUED_2:func.14">&lt;-&gt;</a> <font color="Olive" title="b1">f1</font></span><br/><b>let </b><font color="Maroon" title="c7">f1</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c5">Y1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font> being <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="valued_2.html#K24" title="VALUED_2:func.24">&lt;#&gt;</a> <font color="Olive" title="b1">g</font> holds <br/><font color="Maroon" title="c6">f</font> <a href="valued_2.html#K24" title="VALUED_2:func.24">&lt;#&gt;</a> <span class="p1">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_2.html#K14" title="VALUED_2:func.14">&lt;-&gt;</a> <font color="Maroon" title="c7">f1</font></span><br/><b>let </b><font color="Maroon" title="c8">g</font> be  <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="valued_2.html#K24" title="VALUED_2:func.24">&lt;#&gt;</a> <font color="Maroon" title="c8">g</font> implies <font color="Maroon" title="c6">f</font> <a href="valued_2.html#K24" title="VALUED_2:func.24">&lt;#&gt;</a> <span class="p1">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c8">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_2.html#K14" title="VALUED_2:func.14">&lt;-&gt;</a> <font color="Maroon" title="c7">f1</font> )</span><br/>













<b>assume </b><a NAME="E1:203"/><i><font color="Green" title="E64">A1</font></i>: 
<font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="valued_2.html#K24" title="VALUED_2:func.24">&lt;#&gt;</a> <font color="Maroon" title="c8">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">f</font> <a href="valued_2.html#K24" title="VALUED_2:func.24">&lt;#&gt;</a> <span class="p1">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c8">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_2.html#K14" title="VALUED_2:func.14">&lt;-&gt;</a> <font color="Maroon" title="c7">f1</font></span><br/>

<a NAME="E2:203"/><b>then </b><i><font color="Green" title="E65">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="valued_2.html#D42" target="_self" title="VALUED_2:def.42">Def42</a></i>;<br/>
<a NAME="E3:203"/><i><font color="Green" title="E66">A9</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="valued_2.html#K24" title="VALUED_2:func.24">&lt;#&gt;</a> <span class="p2">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c8">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c8">g</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="valued_2.html#D42" target="_self" title="VALUED_2:def.42">Def42</a></i>;<br/>
<a NAME="E4:203"/><i><font color="Green" title="E67">A4</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="valued_2.html#K14" title="VALUED_2:func.14">&lt;-&gt;</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f1</font>
 
<b>by </b><i><a class="ref" href="valued_2.html#D32" target="_self" title="VALUED_2:def.32">Def32</a></i>;<br/>
<b>hence </b><a NAME="E5:203"/><i><font color="Green" title="E68">A5</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="valued_2.html#K24" title="VALUED_2:func.24">&lt;#&gt;</a> <span class="p2">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c8">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="valued_2.html#K14" title="VALUED_2:func.14">&lt;-&gt;</a> <font color="Maroon" title="c7">f1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:203"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="#E3:203"><i><font color="Green" title="E66">A9</font></i></a>, <a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D17" title="FUNCT_1:def.17">FUNCT_1:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="valued_2.html#K24" title="VALUED_2:func.24">&lt;#&gt;</a> <span class="p2">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c8">g</font></span>)</span></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="valued_2.html#K24" title="VALUED_2:func.24">&lt;#&gt;</a> <span class="p2">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c8">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valued_2.html#K14" title="VALUED_2:func.14">&lt;-&gt;</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="valued_2.html#K24" title="VALUED_2:func.24">&lt;#&gt;</a> <span class="p2">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c8">g</font></span>)</span></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="valued_2.html#K24" title="VALUED_2:func.24">&lt;#&gt;</a> <span class="p2">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c8">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valued_2.html#K14" title="VALUED_2:func.14">&lt;-&gt;</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )</span><br/>

<b>assume </b><a NAME="E6:203"/><i><font color="Green" title="E69">A6</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="valued_2.html#K24" title="VALUED_2:func.24">&lt;#&gt;</a> <span class="p2">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c8">g</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="valued_2.html#K24" title="VALUED_2:func.24">&lt;#&gt;</a> <span class="p2">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c8">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valued_2.html#K14" title="VALUED_2:func.14">&lt;-&gt;</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span><br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="valued_2.html#K24" title="VALUED_2:func.24">&lt;#&gt;</a> <span class="p2">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c8">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="valued_1.html#NK26" title="VALUED_1:NK.26">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c8">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E6:203"><i><font color="Green" title="E69">A6</font></i></a>, <a class="ref" href="valued_2.html#D42" target="_self" title="VALUED_2:def.42">Def42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="valued_1.html#NK26" title="VALUED_1:NK.26">(#)</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></i>
<br/>.= 
 <a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="valued_1.html#NK26" title="VALUED_1:NK.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="valued_2.html#T25" target="_self" title="VALUED_2:th.25">Th19</a></i>
<br/>.= 
 <a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:203"><i><font color="Green" title="E64">A1</font></i></a>, <a class="txt" href="#E6:203"><i><font color="Green" title="E69">A6</font></i></a>, <a class="txt" href="#E4:203"><i><font color="Green" title="E67">A4</font></i></a>, <a class="txt" href="#E5:203"><i><font color="Green" title="E68">A5</font></i></a>, <a class="ref" href="valued_2.html#D42" target="_self" title="VALUED_2:def.42">Def42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="valued_2.html#K14" title="VALUED_2:func.14">&lt;-&gt;</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
<b>by </b><i><a class="txt" href="#E6:203"><i><font color="Green" title="E69">A6</font></i></a>, <a class="txt" href="#E5:203"><i><font color="Green" title="E68">A5</font></i></a>, <a class="ref" href="valued_2.html#D32" target="_self" title="VALUED_2:def.32">Def32</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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