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

<b>let </b><font color="Maroon">V</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of  <a href="complfld.html#K1">F_Complex</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="hahnban1.html#NM1">Functional</a> of <font color="Maroon">V</font>;<br/>



<b>thus </b><a NAME="E1:37"/>
 <a href="vectsp10.html#K8">ker</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="vectsp10.html#K8">ker</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K8">ker</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:37_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  )
 ;<br/>
<a NAME="E4:37_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="ref" href="hermitan.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>hence </b><a NAME="E5:37_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K8">ker</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span>
 <b>by </b><i><a class="ref" href="hermitan.html#T1" target="_self">Th1</a></i>;<br/>


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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:37"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K8">ker</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:37"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  )
 ;<br/>
<a NAME="E5:37"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="ref" href="hermitan.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>hence </b><a NAME="E6:37"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K8">ker</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="hermitan.html#T1" target="_self">Th1</a></i>;<br/>


</div>
