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

<b>consider </b><font color="Maroon">A</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:33"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font> )
 <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">V</font> =  <a href="cfuncdom.html#K6" target="_self">ComplexVectSpace</a> <font color="Maroon">A</font>;<br/>

<b>consider </b><font color="Maroon">f</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K2">COMPLEX</a> , <font color="Maroon">g</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E4:33"/><i><font color="Green">E37</font></i>: 
( ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="cfunct_1.html#NM1">Complex</a>  st <span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">A</font> holds <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a>  &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a>  ) ) &amp; ( for <font color="Olive">h</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K2">COMPLEX</a>   ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="cfunct_1.html#NM1">Complex</a> st <font color="Olive">h</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> ) )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">f</font>, <font color="Maroon">v</font> = <font color="Maroon">g</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> ;<br/>
<b>take </b>
<font color="Maroon">u</font>
;<br/>

<b>take </b>
<font color="Maroon">v</font>
;<br/>

<b>thus </b><a NAME="E6:33"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="cfunct_1.html#NM1">Complex</a>  st <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> holds <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="cfunct_1.html#NM1">Complex</a>, <font color="Maroon">s</font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/>

<b>assume </b><a NAME="E1:33_1"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">r</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">u</font>, <font color="Maroon">v'</font> = <font color="Maroon">s</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K2">COMPLEX</a>  ;<br/>
<a NAME="E3:33_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:33_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="cfuncdom.html#K4" target="_self">ComplexFuncZero</a> <font color="Maroon">A</font>
 
;<br/>
<b>hence </b><a NAME="E5:33_1"/>
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="cfuncdom.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E7:33"/>
for <font color="Olive">w</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="cfunct_1.html#NM1">Complex</a> st <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>

<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">w</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K2">COMPLEX</a>  ;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="cfunct_1.html#NM1">Complex</a>, <font color="Maroon">b</font> being   <a href="cfunct_1.html#NM1">Complex</a><b> such that </b><br/><a NAME="E3:33_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="cfuncdom.html#K1" target="_self">ComplexFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cfuncdom.html#K3" target="_self">ComplexFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="cfuncdom.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:33_2"/>
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:33_2"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="cfunct_1.html#NM1">Complex</a> st <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>


</div>
