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

<b>set </b><font color="Maroon">vq</font> =  <a href="vectsp10.html#K6" target="_self">VectQuot</a> <font color="Maroon">V</font>,<font color="Maroon">W</font>;<br/>
<b>set </b><font color="Maroon">C</font> =  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font>;<br/>
<b>set </b><font color="Maroon">aC</font> =  <a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">V</font>,<font color="Maroon">W</font>;<br/>
<a NAME="E1:68_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" target="_self">VectQuot</a> <font color="Maroon">V</font>,<font color="Maroon">W</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">a</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font>;<br/>
<div><i><font color="Green">E42</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">A</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" target="_self">VectQuot</a> <font color="Maroon">V</font>,<font color="Maroon">W</font></span>)</span>;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E2:68_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>;<br/><b>thus </b><a NAME="E3:68_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>,<font color="Maroon">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">a1</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/>

<b>assume </b><a NAME="E1:68_1_1_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E2:68_1_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a1</font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_4.html#T59">VECTSP_4:59</a></i>;<br/>
<b>then consider </b><font color="Maroon">w</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:68_1_1_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">w</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font> )
 <b>by </b><i><a class="ref" href="vectsp_4.html#T57">VECTSP_4:57</a></i>;<br/>
<a NAME="E5:68_1_1_1_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E6:68_1_1_1_1"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K8" target="_self">ker</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="vectsp10.html#E1:68_1_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">aa</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E8:68_1_1_1_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">aa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">aa</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> )
 ;<br/>
<b>thus </b><font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i>, <a class="ref" href="hahnban1.html#D11">HAHNBAN1:def 11</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>
;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>consider </b><font color="Maroon">ff</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" target="_self">VectQuot</a> <font color="Maroon">V</font>,<font color="Maroon">W</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E4:68_1_1"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">A</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" target="_self">VectQuot</a> <font color="Maroon">V</font>,<font color="Maroon">W</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>,<font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="ref" href="vectsp10.html#T2" target="_self">Th2</a>);<br/></i>
<b>reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">ff</font> as   <a href="hahnban1.html#NM1">Functional</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" target="_self">VectQuot</a> <font color="Maroon">V</font>,<font color="Maroon">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">A</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" target="_self">VectQuot</a> <font color="Maroon">V</font>,<font color="Maroon">W</font></span>)</span>, <font color="Maroon">B</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" target="_self">VectQuot</a> <font color="Maroon">V</font>,<font color="Maroon">W</font></span>)</span>;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E2:68_1_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">b</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:68_1_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/><a NAME="E5:68_1_1_2"/><i><font color="Green">E76</font></i>: 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" target="_self">VectQuot</a> <font color="Maroon">V</font>,<font color="Maroon">W</font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">V</font>,<font color="Maroon">W</font>
 <b>by </b><i/>;<br/><a NAME="E6:68_1_1_2"/><i><font color="Green">E100</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">V</font>,<font color="Maroon">W</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font>
 <b>by </b><i>, , , </i>;<br/><b>thus </b><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">B</font></span>)</span> = 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="vectsp10.html#E2:68_1_1"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D11">HAHNBAN1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">ff</font> as  <a href="hahnban1.html#V1">additive</a> <a href="hahnban1.html#NM1">Functional</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" target="_self">VectQuot</a> <font color="Maroon">V</font>,<font color="Maroon">W</font></span>)</span> <b>by </b><i><a class="ref" href="hahnban1.html#D11">HAHNBAN1:def 11</a></i>;<br/>
<b>take </b>
<font color="Maroon">ff</font>
;<br/>

<b>thus </b><a NAME="E8:68_1_1"/>
for <font color="Olive">A</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" target="_self">VectQuot</a> <font color="Maroon">V</font>,<font color="Maroon">W</font></span>)</span><br/> for <font color="Olive">a</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>  st <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font> holds <br/><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font>
 <b>by </b><i/>;<br/>


</div>
