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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>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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">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> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</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"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">b</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="E20">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</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="E1:34_1_1_1"/>
<font color="Maroon" title="c5">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="c7">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="E3:34_1_1_1"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c7">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font>
 ;<br/><a NAME="E4:34_1_1_1"/>
<font color="Maroon" title="c6">B</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">B1</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="E6:34_1_1_1"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c8">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</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="E8:34_1_1_1"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c7">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>consider </b><font color="Maroon" title="c10">b</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="E10:34_1_1_1"/><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c8">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">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="vectsp_4.html#D6" title="VECTSP_4:def.6">VECTSP_4:def 6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">z</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c10">b</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="E12:34_1_1_1"/>
<font color="Maroon" title="c11">z</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="c12">z</font> = <font color="Maroon" title="c11">z</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="c13">z</font> = <font color="Maroon" title="c12">z</font>;<br/><b>thus </b><a NAME="E14:34_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c13">z</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">a1</font>, <font color="Maroon" title="c15">b1</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="E1:34_1_1_1_1"/>
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a1</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c3">W</font> &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b1</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c3">W</font> )
 ;<br/>

<a NAME="E2:34_1_1_1_1"/><b>then </b><i><font color="Green" title="E25">A6</font></i>: 
( <font color="Maroon" title="c14">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> &amp; <font color="Maroon" title="c15">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">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="txt" href="vectsp10.html#E3:34_1_1_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="vectsp10.html#E6:34_1_1_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="vectsp10.html#E8:34_1_1_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="vectsp10.html#E10:34_1_1_1"><i><font color="Green" title="E24">A5</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="c16">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="E4:34_1_1_1_1"/><i><font color="Green" title="E26">A7</font></i>: 
( <font color="Maroon" title="c16">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="c14">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="c16">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/>
<b>consider </b><font color="Maroon" title="c17">w2</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="E6:34_1_1_1_1"/><i><font color="Green" title="E27">A8</font></i>: 
( <font color="Maroon" title="c17">w2</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="c15">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c17">w2</font> )
 <b>by </b><i><a class="txt" href="vectsp10.html#E2:34_1_1_1_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="vectsp_4.html#T57" title="VECTSP_4:th.57">VECTSP_4:57</a></i>;<br/>
<i><font color="Green" title="E28">A9</font></i>: <font color="Maroon" title="c14">a1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c15">b1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">w1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c17">w2</font>
<b>by </b><i><a class="txt" href="vectsp10.html#E4:34_1_1_1_1"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="vectsp10.html#E6:34_1_1_1_1"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">w1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c17">w2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c10">b</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="c16">w1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c17">w2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
;<br/>
<a NAME="E8:34_1_1_1_1"/>
<font color="Maroon" title="c16">w1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c17">w2</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#E4:34_1_1_1_1"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="vectsp10.html#E6:34_1_1_1_1"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="vectsp_4.html#T28" title="VECTSP_4:th.28">VECTSP_4:28</a></i>;<br/>
<a NAME="E9:34_1_1_1_1"/><b>then </b><i><font color="Green" title="E29">A10</font></i>: 
<font color="Maroon" title="c14">a1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c15">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c10">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#E7:34_1_1_1_1"><i><font color="Green" title="E28">A9</font></i></a></i>;<br/>
<a NAME="E10:34_1_1_1_1"/>
<font color="Maroon" title="c14">a1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c15">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">a1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c15">b1</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="E11:34_1_1_1_1"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">a1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c15">b1</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#E9:34_1_1_1_1"><i><font color="Green" title="E29">A10</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="c5">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"><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 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="E3:34_1_1"/><i><font color="Green" title="E21">A11</font></i>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</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">A</font>,<font color="Olive" title="b2">B</font>,<font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</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#E1:34_1_1"><i><font color="Green" title="E20">A1</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c5">f</font> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</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="c6">f</font>
;<br/>

<b>let </b><font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c8">B</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>, <font color="Maroon" title="c10">b</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="E5:34_1_1"/><i><font color="Green" title="E22">A12</font></i>: 
( <font color="Maroon" title="c7">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> &amp; <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</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="E6:34_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">A</font>,<font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c10">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#E3:34_1_1"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="vectsp10.html#E5:34_1_1"><i><font color="Green" title="E22">A12</font></i></a></i>;<br/>


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