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

<b>set </b><font color="Maroon" title="c4">qf</font> =  <a href="vectsp10.html#K11" title="VECTSP10:func.11">CQFunctional</a> <font color="Maroon" title="c3">f</font>;<br/>
<b>set </b><font color="Maroon" title="c5">W</font> =  <a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font>;<br/>
<b>set </b><font color="Maroon" title="c6">qV</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p1">(<span class="default"><a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font></span>)</span>;<br/>
<a NAME="E1:73_1_1"/><i><font color="Green" title="E38">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> <font color="Maroon" title="c2">V</font>,<span class="p1">(<span class="default"><a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp10.html#D6" target="_self" title="VECTSP10:def.6">Def6</a></i>;<br/>
<a NAME="E2:73_1_1"/><i><font color="Green" title="E39">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="vectsp10.html#D11" target="_self" title="VECTSP10:def.11">Def11</a></i>;<br/>
<b>thus </b><a NAME="E3:73_1_1"/>
 <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K11" title="VECTSP10:func.11">CQFunctional</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p4">(<span class="default"><a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>,<a class="ref" href="vectsp10.html#D10" target="_self" title="VECTSP10:def.10">VECTSP10:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p4">(<span class="default"><a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K11" title="VECTSP10:func.11">CQFunctional</a> <font color="Maroon" title="c3">f</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K11" title="VECTSP10:func.11">CQFunctional</a> <font color="Maroon" title="c3">f</font></span>)</span> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p4">(<span class="default"><a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>

<b>assume </b><a NAME="E1:73_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K11" title="VECTSP10:func.11">CQFunctional</a> <font color="Maroon" title="c3">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p4">(<span class="default"><a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<b>then consider </b><font color="Maroon" title="c8">w</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:73_1_1_1"/><i><font color="Green" title="E40">A3</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w</font> &amp; <span class="p1">(<span class="default"><a href="vectsp10.html#K11" title="VECTSP10:func.11">CQFunctional</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> )
 ;<br/>
<a NAME="E4:73_1_1_1"/>
<font color="Maroon" title="c8">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> <font color="Maroon" title="c2">V</font>,<span class="p1">(<span class="default"><a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp10.html#E1:73_1_1"><i><font color="Green" title="E38">A1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">A</font> being    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of  <a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font><b> such that </b><br/><a NAME="E6:73_1_1_1"/><i><font color="Green" title="E41">A4</font></i>: 
<font color="Maroon" title="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c10">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E8:73_1_1_1"/><i><font color="Green" title="E42">A5</font></i>: 
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D6" title="VECTSP_4:def.6">VECTSP_4:def 6</a></i>;<br/>
<a NAME="E9:73_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 
<b>by </b><i><a class="txt" href="vectsp10.html#E2:73_1_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="vectsp10.html#E3:73_1_1_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="vectsp10.html#E6:73_1_1_1"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="vectsp10.html#E8:73_1_1_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="vectsp10.html#D12" target="_self" title="VECTSP10:def.12">Def12</a></i>;<br/>
<a NAME="E10:73_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">v</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/>
<a NAME="E11:73_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="vectsp10.html#E2:73_1_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>then </b><font color="Maroon" title="c8">w</font> = 
 <a href="vectsp10.html#K4" title="VECTSP10:func.4">zeroCoset</a> <font color="Maroon" title="c2">V</font>,<span class="p1">(<span class="default"><a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp10.html#E6:73_1_1_1"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="vectsp10.html#E8:73_1_1_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="vectsp_4.html#T64" title="VECTSP_4:th.64">VECTSP_4:64</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp10.html#T22" target="_self" title="VECTSP10:th.22">Th22</a></i>
;<br/>
<b>hence </b><a NAME="E13:73_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p4">(<span class="default"><a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></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#E3:73_1_1_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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><b>end;</b></div>

<b>thus </b><a NAME="E4:73_1_1"/>
<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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p4">(<span class="default"><a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K11" title="VECTSP10:func.11">CQFunctional</a> <font color="Maroon" title="c3">f</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p4">(<span class="default"><a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K11" title="VECTSP10:func.11">CQFunctional</a> <font color="Maroon" title="c3">f</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:73_1_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p4">(<span class="default"><a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K11" title="VECTSP10:func.11">CQFunctional</a> <font color="Maroon" title="c3">f</font></span>)</span></span><br/>

<a NAME="E2:73_1_1_2"/><b>then </b><i><font color="Green" title="E40">A6</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:73_1_1_2"/>
<span class="p1">(<span class="default"><a href="vectsp10.html#K11" title="VECTSP10:func.11">CQFunctional</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p3">(<span class="default"><a href="vectsp10.html#K9" title="VECTSP10:func.9">Ker</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 
<b>by </b><i><a class="ref" href="hahnban1.html#D13" title="HAHNBAN1:def.13">HAHNBAN1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E4:73_1_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K11" title="VECTSP10:func.11">CQFunctional</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp10.html#E2:73_1_1_2"><i><font color="Green" title="E40">A6</font></i></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><b>end;</b></div>


</div>
