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<b>let </b><font color="Maroon" title="c2">c1</font>, <font color="Maroon" title="c3">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">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">g</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">g</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> &amp; <font color="Olive" title="b1">g</font> <a href="valued_1.html#NK26" title="VALUED_1:NK.26">(#)</a> <font color="Maroon" title="c2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="valued_1.html#NK26" title="VALUED_1:NK.26">(#)</a> <font color="Maroon" title="c3">c2</font> holds <br/><font color="Maroon" title="c2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font></span><br/><b>let </b><font color="Maroon" title="c4">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="c4">g</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="c4">g</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> &amp; <font color="Maroon" title="c4">g</font> <a href="valued_1.html#NK26" title="VALUED_1:NK.26">(#)</a> <font color="Maroon" title="c2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="valued_1.html#NK26" title="VALUED_1:NK.26">(#)</a> <font color="Maroon" title="c3">c2</font> implies <font color="Maroon" title="c2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:90"/><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c4">g</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="E2:90"/><i><font color="Green" title="E36">A4</font></i>: 
<font color="Maroon" title="c4">g</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 <b>and </b><br/><a NAME="E3:90"/><i><font color="Green" title="E37">A0</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="valued_1.html#NK26" title="VALUED_1:NK.26">(#)</a> <font color="Maroon" title="c2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="valued_1.html#NK26" title="VALUED_1:NK.26">(#)</a> <font color="Maroon" title="c3">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="c2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font></span><br/>

<b>consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:90"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c5">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> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="#E1:90"><i><font color="Green" title="E35">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="E6:90"/>
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="txt" href="#E5:90"><i><font color="Green" title="E38">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="E7:90"/><b>then </b><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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="#E2:90"><i><font color="Green" title="E36">A4</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="E8:90"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="valued_1.html#NK26" title="VALUED_1:NK.26">(#)</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">c1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="valued_1.html#NK26" title="VALUED_1:NK.26">(#)</a> <font color="Maroon" title="c3">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">c2</font> )
 
<b>by </b><i><a class="ref" href="valued_1.html#T6" title="VALUED_1:th.6">VALUED_1:6</a></i>;<br/>
<b>hence </b><a NAME="E9:90"/>
<font color="Maroon" title="c2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font>
 <b>by </b><i><a class="txt" href="#E3:90"><i><font color="Green" title="E37">A0</font></i></a>, <a class="txt" href="#E7:90"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</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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