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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">c1</font>, <font color="Olive" title="b4">c2</font> being <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/> for <font color="Olive" title="b5">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="b6">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>  st <font color="Olive" title="b6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <font color="Olive" title="b3">c1</font> holds <br/><font color="Olive" title="b6">f1</font> <a href="valued_2.html#K18" title="VALUED_2:func.18">[+]</a> <font color="Olive" title="b4">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">c1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">c2</font></span>)</span></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">c1</font>, <font color="Olive" title="b2">c2</font> being <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="Maroon" title="c4">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="Maroon" title="c5">Y1</font>  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#K19" title="VALUED_2:func.19">[-]</a> <font color="Olive" title="b1">c1</font> holds <br/><font color="Olive" title="b4">f1</font> <a href="valued_2.html#K18" title="VALUED_2:func.18">[+]</a> <font color="Olive" title="b2">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">c1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">c2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c6">c1</font>, <font color="Maroon" title="c7">c2</font> be  <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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>  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#K19" title="VALUED_2:func.19">[-]</a> <font color="Maroon" title="c6">c1</font> holds <br/><font color="Olive" title="b2">f1</font> <a href="valued_2.html#K18" title="VALUED_2:func.18">[+]</a> <font color="Maroon" title="c7">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">c2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c8">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>  st <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <font color="Maroon" title="c6">c1</font> holds <br/><font color="Olive" title="b1">f1</font> <a href="valued_2.html#K18" title="VALUED_2:func.18">[+]</a> <font color="Maroon" title="c7">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">c2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c9">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"> ( <font color="Maroon" title="c9">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <font color="Maroon" title="c6">c1</font> implies <font color="Maroon" title="c9">f1</font> <a href="valued_2.html#K18" title="VALUED_2:func.18">[+]</a> <font color="Maroon" title="c7">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">c2</font></span>)</span> )</span><br/>















<b>assume </b><a NAME="E1:161"/><i><font color="Green" title="E59">A1</font></i>: 
<font color="Maroon" title="c9">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <font color="Maroon" title="c6">c1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">f1</font> <a href="valued_2.html#K18" title="VALUED_2:func.18">[+]</a> <font color="Maroon" title="c7">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">c2</font></span>)</span></span><br/>

<a NAME="E2:161"/><b>then </b><i><font color="Green" title="E60">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f1</font> <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="c8">f</font>
 
<b>by </b><i><a class="ref" href="valued_2.html#D36" target="_self" title="VALUED_2:def.36">Def36</a></i>;<br/>
<a NAME="E3:161"/><i><font color="Green" title="E61">A3</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="c9">f1</font> <a href="valued_2.html#K18" title="VALUED_2:func.18">[+]</a> <font color="Maroon" title="c7">c2</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="c9">f1</font>
 
<b>by </b><i><a class="ref" href="valued_2.html#D36" target="_self" title="VALUED_2:def.36">Def36</a></i>;<br/>
<b>hence </b><a NAME="E4:161"/><i><font color="Green" title="E62">A4</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="c9">f1</font> <a href="valued_2.html#K18" title="VALUED_2:func.18">[+]</a> <font color="Maroon" title="c7">c2</font></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"><font color="Maroon" title="c8">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">c1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">c2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:161"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="valued_2.html#D36" target="_self" title="VALUED_2:def.36">Def36</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="c9">f1</font> <a href="valued_2.html#K18" title="VALUED_2:func.18">[+]</a> <font color="Maroon" title="c7">c2</font></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="valued_2.html#K18" title="VALUED_2:func.18">[+]</a> <font color="Maroon" title="c7">c2</font></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"><font color="Maroon" title="c8">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">c1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">c2</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> )</span><br/>

<b>let </b><font color="Maroon" title="c10">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="c10">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="c9">f1</font> <a href="valued_2.html#K18" title="VALUED_2:func.18">[+]</a> <font color="Maroon" title="c7">c2</font></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="valued_2.html#K18" title="VALUED_2:func.18">[+]</a> <font color="Maroon" title="c7">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">c1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">c2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> )</span><br/>

<b>assume </b><a NAME="E5:161"/><i><font color="Green" title="E63">A5</font></i>: 
<font color="Maroon" title="c10">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="c9">f1</font> <a href="valued_2.html#K18" title="VALUED_2:func.18">[+]</a> <font color="Maroon" title="c7">c2</font></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="c9">f1</font> <a href="valued_2.html#K18" title="VALUED_2:func.18">[+]</a> <font color="Maroon" title="c7">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">c1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">c2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span><br/>

<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="valued_2.html#K18" title="VALUED_2:func.18">[+]</a> <font color="Maroon" title="c7">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="valued_1.html#NK8" title="VALUED_1:NK.8">+</a> <font color="Maroon" title="c7">c2</font>
<b>by </b><i><a class="ref" href="valued_2.html#D36" target="_self" title="VALUED_2:def.36">Def36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="valued_1.html#K13" title="VALUED_1:func.13">-</a> <font color="Maroon" title="c6">c1</font></span>)</span> <a href="valued_1.html#NK8" title="VALUED_1:NK.8">+</a> <font color="Maroon" title="c7">c2</font>
<b>by </b><i><a class="txt" href="#E1:161"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="#E3:161"><i><font color="Green" title="E61">A3</font></i></a>, <a class="txt" href="#E5:161"><i><font color="Green" title="E63">A5</font></i></a>, <a class="ref" href="valued_2.html#D36" target="_self" title="VALUED_2:def.36">Def36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="valued_1.html#K13" title="VALUED_1:func.13">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">c2</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_2.html#T15" target="_self" title="VALUED_2:th.15">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">c1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">c2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="#E5:161"><i><font color="Green" title="E63">A5</font></i></a>, <a class="txt" href="#E4:161"><i><font color="Green" title="E62">A4</font></i></a>, <a class="ref" href="valued_2.html#D36" target="_self" title="VALUED_2:def.36">Def36</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
