<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c2">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"> for <font color="Olive" title="b1">Y</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="b2">c1</font>, <font color="Olive" title="b3">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="b4">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>  st <font color="Olive" title="b4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b4">f</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> &amp; <font color="Olive" title="b4">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <font color="Olive" title="b2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <font color="Olive" title="b3">c2</font> holds <br/><font color="Olive" title="b2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c2</font></span><br/><b>let </b><font color="Maroon" title="c3">Y</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="c3">Y</font>  st <font color="Olive" title="b3">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b3">f</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> &amp; <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> <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="b2">c2</font> holds <br/><font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">c2</font></span><br/><b>let </b><font color="Maroon" title="c4">c1</font>, <font color="Maroon" title="c5">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="c3">Y</font>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">f</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> &amp; <font color="Olive" title="b1">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <font color="Maroon" title="c4">c1</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="c5">c2</font> holds <br/><font color="Maroon" title="c4">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c2</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="c3">Y</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="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c6">f</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> &amp; <font color="Maroon" title="c6">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <font color="Maroon" title="c4">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <font color="Maroon" title="c5">c2</font> implies <font color="Maroon" title="c4">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c2</font> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:159"/><i><font color="Green" title="E59">A2</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E2:159"/><i><font color="Green" title="E60">A5</font></i>: 
<font color="Maroon" title="c6">f</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 <b>and </b><br/><a NAME="E3:159"/><i><font color="Green" title="E61">A0</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <font color="Maroon" title="c4">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <font color="Maroon" title="c5">c2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c2</font></span><br/>

<b>consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:159"/><i><font color="Green" title="E62">A1</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="#E1:159"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E6:159"/>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="#E5:159"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:159"/><b>then </b><i><font color="Green" title="E63">A4</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="#E2:159"><i><font color="Green" title="E60">A5</font></i></a>, <a class="ref" href="relat_1.html#D9" title="RELAT_1:def.9">RELAT_1:def 9</a></i>;<br/>
<a NAME="E8:159"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> <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="c6">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <font color="Maroon" title="c4">c1</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> <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="c6">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <font color="Maroon" title="c5">c2</font></span>)</span> )
 
<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="E9:159"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c7">x</font></span>)</span> <a href="valued_1.html#K13" title="VALUED_1:func.13">-</a> <font color="Maroon" title="c4">c1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="valued_2.html#K19" title="VALUED_2:func.19">[-]</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c7">x</font></span>)</span> <a href="valued_1.html#K13" title="VALUED_1:func.13">-</a> <font color="Maroon" title="c5">c2</font> )
 
<b>by </b><i><a class="txt" href="#E5:159"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="valued_2.html#D36" target="_self" title="VALUED_2:def.36">Def36</a></i>;<br/>
<b>hence </b><a NAME="E10:159"/>
<font color="Maroon" title="c4">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c2</font>
 <b>by </b><i><a class="txt" href="#E3:159"><i><font color="Green" title="E61">A0</font></i></a>, <a class="txt" href="#E7:159"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="valued_2.html#T9" target="_self" title="VALUED_2:th.9">Thb</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
