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

<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">cF</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</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>2</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>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Olive">a</font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font>;<br/>
<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font>;<br/><a NAME="E1:36_1_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font>
 ;<br/><b>then consider </b><font color="Maroon">A1</font> being    <a href="vectsp_4.html#M2">Coset</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E3:36_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 ;<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="E5:36_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">A1</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><a class="ref" href="vectsp_4.html#D6">VECTSP_4:def 6</a></i>;<br/><b>reconsider </b><font color="Maroon">c</font> = <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a</font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font> as    <a href="vectsp_4.html#M2">Coset</a> of <font color="Maroon">W</font> <b>by </b><i><a class="ref" href="vectsp_4.html#D6">VECTSP_4:def 6</a></i>;<br/><a NAME="E7:36_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font>
 ;<br/><b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font> ;<br/><b>take </b><font color="Maroon">c</font> = <font color="Maroon">c</font>;<br/><b>thus </b><a NAME="E9:36_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">z</font>,<font color="Maroon">A</font>,<font color="Maroon">c</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_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="E1:36_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="E2:36_1_1_1_1"/><b>then </b>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</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">w1</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:36_1_1_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">w1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w1</font> )
 <b>by </b><i><a class="ref" href="vectsp_4.html#T57">VECTSP_4:57</a></i>;<br/>
<a NAME="E5:36_1_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">z</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>;<br/>
<a NAME="E6:36_1_1_1_1"/>
<font color="Maroon">z</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_4.html#T29">VECTSP_4:29</a></i>;<br/>
<a NAME="E7:36_1_1_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">z</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a</font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:36_1_1_1_1"/>
<font color="Maroon">z</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a1</font></span>)</span> <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>hence </b><a NAME="E9:36_1_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a1</font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="vectsp_4.html#T71">VECTSP_4:71</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>,<span class="p2">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font><b> such that </b><br/><a NAME="E3:36_1_1"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font><br/> for <font color="Olive">A</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">z</font>,<font color="Olive">A</font>,<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">z</font>,<font color="Olive">A</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/>





<b>assume </b><a NAME="E4:36_1_1"/><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>
 ;<br/>

<b>thus </b><a NAME="E5:36_1_1"/>
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a</font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font>
 <b>by </b><i>, </i>;<br/>


</div>
