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<b>set </b><font color="Maroon" title="c4">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="c5">cF</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</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#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>  st $2 <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/>$3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">$1 <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c3">W</font>;<br/>
<div><i><font color="Green" title="E21">A1</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" title="c6">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>;<br/><b>let </b><font color="Maroon" title="c7">A</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <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/><a NAME="E1:36_1_1_1"/>
<font color="Maroon" title="c7">A</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>,<font color="Maroon" title="c3">W</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">A1</font> being    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c3">W</font><b> such that </b><br/><a NAME="E3:36_1_1_1"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c8">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font>
 ;<br/><b>consider </b><font color="Maroon" title="c9">a</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="E5:36_1_1_1"/><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c8">A1</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="vectsp_4.html#D6" title="VECTSP_4:def.6">VECTSP_4:def 6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">c</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c3">W</font> as    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c3">W</font> <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="E7:36_1_1_1"/>
<font color="Maroon" title="c10">c</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>,<font color="Maroon" title="c3">W</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c11">c</font> = <font color="Maroon" title="c10">c</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <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>take </b><font color="Maroon" title="c12">c</font> = <font color="Maroon" title="c11">c</font>;<br/><b>thus </b><a NAME="E9:36_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c12">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" title="c13">a1</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>;<br/>

<b>assume </b><a NAME="E1:36_1_1_1_1"/>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a1</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c3">W</font>
 ;<br/>

<a NAME="E2:36_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="txt" href="vectsp10.html#E3:36_1_1_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="vectsp10.html#E5:36_1_1_1"><i><font color="Green" title="E23">A3</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="c14">w1</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="E4:36_1_1_1_1"/><i><font color="Green" title="E24">A4</font></i>: 
( <font color="Maroon" title="c14">w1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W</font> &amp; <font color="Maroon" title="c13">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c14">w1</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="E5:36_1_1_1_1"/><i><font color="Green" title="E25">A5</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c13">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c14">w1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp10.html#E4:36_1_1_1_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>;<br/>
<a NAME="E6:36_1_1_1_1"/>
<font color="Maroon" title="c6">z</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c14">w1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W</font>
 
<b>by </b><i><a class="txt" href="vectsp10.html#E4:36_1_1_1_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="vectsp_4.html#T29" title="VECTSP_4:th.29">VECTSP_4:29</a></i>;<br/>
<a NAME="E7:36_1_1_1_1"/><b>then </b><i><font color="Green" title="E26">A6</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c13">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c9">a</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#E5:36_1_1_1_1"><i><font color="Green" title="E25">A5</font></i></a></i>;<br/>
<a NAME="E8:36_1_1_1_1"/>
<font color="Maroon" title="c6">z</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c13">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c13">a1</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="ref" href="vectsp_4.html#T59" title="VECTSP_4:th.59">VECTSP_4:59</a></i>;<br/>
<b>hence </b><a NAME="E9:36_1_1_1_1"/>
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c13">a1</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#E7:36_1_1_1_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="vectsp_4.html#T71" title="VECTSP_4:th.71">VECTSP_4:71</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><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></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <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><b> such that </b><br/><a NAME="E3:36_1_1"/><i><font color="Green" title="E22">A7</font></i>: 
for <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">A</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <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> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">A</font>,<font color="Maroon" title="c6">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">A</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="vectsp10.html#E1:36_1_1"><i><font color="Green" title="E21">A1</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c6">f</font>
;<br/>

<b>let </b><font color="Maroon" title="c7">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>;<br/><b>let </b><font color="Maroon" title="c8">A</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <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>let </b><font color="Maroon" title="c9">a</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>;<br/>





<b>assume </b><a NAME="E4:36_1_1"/><i><font color="Green" title="E23">A8</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>
 ;<br/>

<b>thus </b><a NAME="E5:36_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">z</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c9">a</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#E3:36_1_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="vectsp10.html#E4:36_1_1"><i><font color="Green" title="E23">A8</font></i></a></i>;<br/>


</div>
