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

<b>set </b><font color="Maroon" title="c5">vq</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>;<br/>
<b>set </b><font color="Maroon" title="c6">C</font> =  <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>;<br/>
<b>set </b><font color="Maroon" title="c7">aC</font> =  <a href="vectsp10.html#K3" title="VECTSP10:func.3">addCoset</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>;<br/>
<a NAME="E1:68_1_1"/><i><font color="Green" title="E36">A2</font></i>: 
 <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</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 <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp10.html#D6" target="_self" title="VECTSP10:def.6">Def6</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">a</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c3">W</font> holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font>;<br/>
<div><i><font color="Green" title="E37">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">A</font> be   <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>,<font color="Maroon" title="c3">W</font></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c9">a</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="E2:68_1_1_1"/><i><font color="Green" title="E38">A4</font></i>: 
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" target="_self" title="VECTSP10:th.23">Th23</a></i>;<br/><b>take </b><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font>;<br/><b>thus </b><a NAME="E3:68_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c10">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">a1</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font>;<br/>

<b>assume </b><a NAME="E1:68_1_1_1_1"/>
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a1</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c3">W</font>
 ;<br/>

<a NAME="E2:68_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">a1</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c3">W</font>
 
<b>by </b><i><a class="txt" href="vectsp10.html#E2:68_1_1_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="vectsp_4.html#T59" title="VECTSP_4:th.59">VECTSP_4:59</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">w</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="E4:68_1_1_1_1"/><i><font color="Green" title="E39">A5</font></i>: 
( <font color="Maroon" title="c12">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font> &amp; <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">w</font> )
 <b>by </b><i><a class="ref" href="vectsp_4.html#T57" title="VECTSP_4:th.57">VECTSP_4:57</a></i>;<br/>
<a NAME="E5:68_1_1_1_1"/>
<font color="Maroon" title="c12">w</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="c3">W</font>
 
<b>by </b><i><a class="txt" href="vectsp10.html#E4:68_1_1_1_1"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E6:68_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">w</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="c4">f</font>
 
<b>by </b><i><a class="txt" href="vectsp10.html#E1:68"><i><font color="Green" title="E35">A1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">aa</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="E8:68_1_1_1_1"/><i><font color="Green" title="E40">A6</font></i>: 
( <font color="Maroon" title="c13">aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">aa</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/>
<b>thus </b><font color="Maroon" title="c10">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">w</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp10.html#E4:68_1_1_1_1"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="hahnban1.html#D11" title="HAHNBAN1:def.11">HAHNBAN1:def 11</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a1</font>
<b>by </b><i><a class="txt" href="vectsp10.html#E8:68_1_1_1_1"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c8">ff</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 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>,<font color="Maroon" title="c3">W</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font><b> such that </b><br/><a NAME="E4:68_1_1"/><i><font color="Green" title="E38">A7</font></i>: 
for <font color="Olive" title="b1">A</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>,<font color="Maroon" title="c3">W</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>,<font color="Maroon" title="c8">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="vectsp10.html#E2:68_1_1"><i><font color="Green" title="E37">A3</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c9">ff</font> = <font color="Maroon" title="c8">ff</font> as   <a href="hahnban1.html#NM1" title="HAHNBAN1:NM.1">Functional</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>,<font color="Maroon" title="c3">W</font></span>)</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">A</font>, <font color="Maroon" title="c11">B</font> be   <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>,<font color="Maroon" title="c3">W</font></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c12">a</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="E2:68_1_1_2"/><i><font color="Green" title="E39">A8</font></i>: 
<font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" target="_self" title="VECTSP10:th.23">Th23</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">b</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="E4:68_1_1_2"/><i><font color="Green" title="E40">A9</font></i>: 
<font color="Maroon" title="c11">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" target="_self" title="VECTSP10:th.23">Th23</a></i>;<br/><a NAME="E5:68_1_1_2"/><i><font color="Green" title="E41">A10</font></i>: 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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>,<font color="Maroon" title="c3">W</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="vectsp10.html#K3" title="VECTSP10:func.3">addCoset</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="ref" href="vectsp10.html#D6" target="_self" title="VECTSP10:def.6">Def6</a></i>;<br/><a NAME="E6:68_1_1_2"/>
<span class="p1">(<span class="default"><a href="vectsp10.html#K3" title="VECTSP10:func.3">addCoset</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">A</font>,<font color="Maroon" title="c11">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="txt" href="vectsp10.html#E1:68_1_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="vectsp10.html#E2:68_1_1_2"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="vectsp10.html#E4:68_1_1_2"><i><font color="Green" title="E40">A9</font></i></a>, <a class="ref" href="vectsp10.html#D3" target="_self" title="VECTSP10:def.3">Def3</a></i>;<br/><b>hence </b><font color="Maroon" title="c9">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">A</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">B</font></span>)</span> = 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">b</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp10.html#E4:68_1_1"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="vectsp10.html#E5:68_1_1_2"><i><font color="Green" title="E41">A10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D11" title="HAHNBAN1:def.11">HAHNBAN1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">A</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp10.html#E4:68_1_1"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="vectsp10.html#E2:68_1_1_2"><i><font color="Green" title="E39">A8</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">A</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">B</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp10.html#E4:68_1_1"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="vectsp10.html#E4:68_1_1_2"><i><font color="Green" title="E40">A9</font></i></a></i>
;<br/><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">ff</font> = <font color="Maroon" title="c9">ff</font> as  <a href="hahnban1.html#V1" title="HAHNBAN1:attr.1">additive</a> <a href="hahnban1.html#NM1" title="HAHNBAN1:NM.1">Functional</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>,<font color="Maroon" title="c3">W</font></span>)</span> <b>by </b><i><a class="ref" href="hahnban1.html#D11" title="HAHNBAN1:def.11">HAHNBAN1:def 11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">ff</font>
;<br/>

<b>thus </b><a NAME="E8:68_1_1"/>
for <font color="Olive" title="b1">A</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>,<font color="Maroon" title="c3">W</font></span>)</span><br/> for <font color="Olive" title="b2">a</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c3">W</font> holds <br/><font color="Maroon" title="c10">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font>
 <b>by </b><i><a class="txt" href="vectsp10.html#E4:68_1_1"><i><font color="Green" title="E38">A7</font></i></a></i>;<br/>


</div>
