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<b>let </b><font color="Maroon" title="c3">f</font> be   <a href="hahnban1.html#NM1" title="HAHNBAN1:NM.1">Functional</a> of <font color="Maroon" title="c2">V</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:66_1_1"/><i><font color="Green" title="E34">A1</font></i>: 
not <font color="Maroon" title="c3">f</font> is <a href="vectsp10.html#V2" title="VECTSP10:attr.2">degenerated</a>
 <b>and </b><br/><a NAME="E2:66_1_1"/><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="hahnban1.html#V3" title="HAHNBAN1:attr.3">0-preserving</a>
 <b>and </b><br/><a NAME="E3:66_1_1"/><i><font color="Green" title="E36">A3</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 ;<br/>

<a NAME="E4:66_1_1"/><i><font color="Green" title="E37">A4</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">K</font>
 
<b>by </b><i><a class="txt" href="vectsp10.html#E2:66_1_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="hahnban1.html#D13" title="HAHNBAN1:def.13">HAHNBAN1:def 13</a></i>;<br/>
<a NAME="E5:66_1_1"/><i><font color="Green" title="E38">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_1_1_1"/><i><font color="Green" title="E39">A6</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">K</font>
 ;<br/><a NAME="E2:66_1_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Maroon" title="c3">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:66_1_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">v</font> = <font color="Maroon" title="c4">x</font> as   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_1_1_1_1"/>
( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font> or <font color="Maroon" title="c5">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:66_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font>
 ;<br/><div class="add"><a NAME="E2:66_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> : <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">K</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="vectsp10.html#E4:66_1_1"><i><font color="Green" title="E37">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E3:66_1_1_1_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Maroon" title="c3">f</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:66_1_1_1_1_1_2"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font>
 ;<br/><div class="add"><a NAME="E2:66_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">K</font>
 <b>by </b><i><a class="txt" href="vectsp10.html#E1:66_1_1_1"><i><font color="Green" title="E39">A6</font></i></a></i>;<br/><a NAME="E3:66_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> : <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">K</font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E4:66_1_1_1_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Maroon" title="c3">f</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then </b>the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> = 
 <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Maroon" title="c3">f</font>
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="txt" href="vectsp10.html#E1:66_1_1"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="vectsp10.html#D10" target="_self" title="VECTSP10:def.10">Def10</a></i>
;<br/><a NAME="E4:66_1_1_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 <b>by </b><i><a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></i>;<br/><b>hence </b><a NAME="E5:66_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="realset2.html#D5" title="REALSET2:def.5">REALSET2:def 5</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c4">v</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E8:66_1_1"/><i><font color="Green" title="E39">A7</font></i>: 
( <font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">K</font> )
 ;<br/>
<b>thus </b><a NAME="E9:66_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="vectsp10.html#E3:66_1_1"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="vectsp10.html#E4:66_1_1"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="vectsp10.html#E5:66_1_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="vectsp10.html#E8:66_1_1"><i><font color="Green" title="E39">A7</font></i></a>, <a class="ref" href="funct_1.html#D16" title="FUNCT_1:def.16">FUNCT_1:def 16</a></i>;<br/>


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