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

<b>set </b><font color="Maroon" title="c3">cR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c4">ccs</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">x</font> </span>}</span> ;<br/>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> = 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>
<br/>.= 
<font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/>
<a NAME="E2:35_1_1"/><b>then </b>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">x</font> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">ccs</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">x</font> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E4:35_1_1"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c5">ccs</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:35_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ccs</font>
 ;<br/>

<a NAME="E2:35_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b1">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Olive" title="b1">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">x'</font> )
 
;<br/>
<b>hence </b><a NAME="E3:35_1_1_2"/>
<font color="Maroon" title="c6">x</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="c1">R</font>
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c6">acs</font> = the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">R</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c5">ccs</font>;<br/>
<b>set </b><font color="Maroon" title="c7">mcs</font> = the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">R</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c5">ccs</font>;<br/>
<b>set </b><font color="Maroon" title="c8">Zs</font> = the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c9">Us</font> =  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font>;<br/>
<a NAME="E5:35_1_1"/><i><font color="Green" title="E30">A2</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c5">ccs</font>,<font color="Maroon" title="c5">ccs</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:35_1_1_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c5">ccs</font>,<font color="Maroon" title="c5">ccs</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">a</font>, <font color="Maroon" title="c12">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:35_1_1_3"/><i><font color="Green" title="E31">A3</font></i>: 
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ccs</font> &amp; <font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ccs</font> )
 <b>and </b><br/><a NAME="E4:35_1_1_3"/><i><font color="Green" title="E32">A4</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>thus </b><a NAME="E5:35_1_1_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:35_1_1"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="weddwitt.html#E3:35_1_1_3"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="weddwitt.html#E4:35_1_1_3"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">acs</font> = the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">R</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c5">ccs</font> as   <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"><font color="Maroon" title="c5">ccs</font>,<font color="Maroon" title="c5">ccs</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="funct_2.html#T38" title="FUNCT_2:th.38">FUNCT_2:38</a></i>;<br/>
<a NAME="E7:35_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">acs</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">ccs</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:35_1_1_4"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">acs</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:35_1_1_4"/><i><font color="Green" title="E31">A5</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">acs</font>
 <b>and </b><br/><a NAME="E4:35_1_1_4"/><i><font color="Green" title="E32">A6</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">acs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c14">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:35_1_1_4"/><i><font color="Green" title="E33">A7</font></i>: 
( <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ccs</font> &amp; <font color="Maroon" title="c14">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ccs</font> )
 <b>and </b><br/><a NAME="E7:35_1_1_4"/><i><font color="Green" title="E34">A8</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:35_1_1_4"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">a</font> = <font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c16">b</font> = <font color="Maroon" title="c14">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="weddwitt.html#E4:35_1_1"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="weddwitt.html#E6:35_1_1_4"><i><font color="Green" title="E33">A7</font></i></a></i>;<br/>
<a NAME="E9:35_1_1_4"/><i><font color="Green" title="E35">A9</font></i>: 
 ex <font color="Olive" title="b1">a'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b1">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a</font> &amp; <font color="Olive" title="b1">a'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">a'</font> )
 
<b>by </b><i><a class="txt" href="weddwitt.html#E6:35_1_1_4"><i><font color="Green" title="E33">A7</font></i></a></i>;<br/>
<a NAME="E10:35_1_1_4"/><i><font color="Green" title="E36">A10</font></i>: 
 ex <font color="Olive" title="b1">b'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b1">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">b</font> &amp; <font color="Olive" title="b1">b'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">b'</font> )
 
<b>by </b><i><a class="txt" href="weddwitt.html#E6:35_1_1_4"><i><font color="Green" title="E33">A7</font></i></a></i>;<br/>
<a NAME="E11:35_1_1_4"/><i><font color="Green" title="E37">A11</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c5">ccs</font>,<font color="Maroon" title="c5">ccs</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E6:35_1_1_4"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E12:35_1_1_4"/><i><font color="Green" title="E38">A12</font></i>: 
<font color="Maroon" title="c15">a</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c16">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">acs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E7:35_1_1_4"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="weddwitt.html#E11:35_1_1_4"><i><font color="Green" title="E37">A11</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c16">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c15">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="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c16">b</font></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E9:35_1_1_4"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="weddwitt.html#E10:35_1_1_4"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="vectsp_1.html#D12" title="VECTSP_1:def.12">VECTSP_1:def 12</a></i>
<br/>.= 
<font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c16">b</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D11" title="VECTSP_1:def.11">VECTSP_1:def 11</a></i>
;<br/>
<b>hence </b><a NAME="E14:35_1_1_4"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ccs</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:35_1_1_4"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="weddwitt.html#E12:35_1_1_4"><i><font color="Green" title="E38">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c11">acs</font> = <font color="Maroon" title="c10">acs</font> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c5">ccs</font> <b>by </b><i><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">mcs</font> = the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">R</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c5">ccs</font> as   <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"><font color="Maroon" title="c5">ccs</font>,<font color="Maroon" title="c5">ccs</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="weddwitt.html#E5:35_1_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="funct_2.html#T38" title="FUNCT_2:th.38">FUNCT_2:38</a></i>;<br/>
<a NAME="E10:35_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">mcs</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">ccs</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:35_1_1_5"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">mcs</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:35_1_1_5"/><i><font color="Green" title="E31">A13</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">mcs</font>
 <b>and </b><br/><a NAME="E4:35_1_1_5"/><i><font color="Green" title="E32">A14</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">mcs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c16">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:35_1_1_5"/><i><font color="Green" title="E33">A15</font></i>: 
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ccs</font> &amp; <font color="Maroon" title="c16">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ccs</font> )
 <b>and </b><br/><a NAME="E7:35_1_1_5"/><i><font color="Green" title="E34">A16</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:35_1_1_5"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">a</font> = <font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c18">b</font> = <font color="Maroon" title="c16">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="weddwitt.html#E4:35_1_1"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="weddwitt.html#E6:35_1_1_5"><i><font color="Green" title="E33">A15</font></i></a></i>;<br/>
<a NAME="E9:35_1_1_5"/><i><font color="Green" title="E35">A17</font></i>: 
 ex <font color="Olive" title="b1">a'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b1">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font> &amp; <font color="Olive" title="b1">a'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">a'</font> )
 
<b>by </b><i><a class="txt" href="weddwitt.html#E6:35_1_1_5"><i><font color="Green" title="E33">A15</font></i></a></i>;<br/>
<a NAME="E10:35_1_1_5"/><i><font color="Green" title="E36">A18</font></i>: 
 ex <font color="Olive" title="b1">b'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b1">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> &amp; <font color="Olive" title="b1">b'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">b'</font> )
 
<b>by </b><i><a class="txt" href="weddwitt.html#E6:35_1_1_5"><i><font color="Green" title="E33">A15</font></i></a></i>;<br/>
<a NAME="E11:35_1_1_5"/><i><font color="Green" title="E37">A19</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c18">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c5">ccs</font>,<font color="Maroon" title="c5">ccs</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E6:35_1_1_5"><i><font color="Green" title="E33">A15</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E12:35_1_1_5"/><i><font color="Green" title="E38">A20</font></i>: 
<font color="Maroon" title="c17">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">mcs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E7:35_1_1_5"><i><font color="Green" title="E34">A16</font></i></a>, <a class="txt" href="weddwitt.html#E11:35_1_1_5"><i><font color="Green" title="E37">A19</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> = 
<font color="Maroon" title="c17">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c18">b</font></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E10:35_1_1_5"><i><font color="Green" title="E36">A18</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c18">b</font>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c18">b</font></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E9:35_1_1_5"><i><font color="Green" title="E35">A17</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E14:35_1_1_5"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ccs</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:35_1_1_5"><i><font color="Green" title="E32">A14</font></i></a>, <a class="txt" href="weddwitt.html#E12:35_1_1_5"><i><font color="Green" title="E38">A20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c13">mcs</font> = <font color="Maroon" title="c12">mcs</font> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c5">ccs</font> <b>by </b><i><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></i>;<br/>
<a NAME="E12:35_1_1"/><i><font color="Green" title="E31">A21</font></i>: 
the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font>
 
;<br/>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> = 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>
<br/>.= 
<font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/>
<a NAME="E14:35_1_1"/><b>then </b>
the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ccs</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">Zs</font> = the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c1">R</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">ccs</font> ;<br/>
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> = 
<font color="Maroon" title="c2">s</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19" title="VECTSP_1:def.19">VECTSP_1:def 19</a></i>
<br/>.= 
<font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>
;<br/>
<a NAME="E17:35_1_1"/><b>then </b>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ccs</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">Us</font> =  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">ccs</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c16">cs</font> =  <a href="vectsp_1.html#G3" title="VECTSP_1:aggr.3">doubleLoopStr</a>(# <font color="Maroon" title="c5">ccs</font>,<font color="Maroon" title="c11">acs</font>,<font color="Maroon" title="c13">mcs</font>,<font color="Maroon" title="c15">Us</font>,<font color="Maroon" title="c14">Zs</font> #) as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a>  ;<br/>
<a NAME="E20:35_1_1"/><i><font color="Green" title="E32">A22</font></i>: 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c16">cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Zs</font>
 
;<br/>
<div><i><font color="Green" title="E33">A23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">x</font>, <font color="Maroon" title="c18">e</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">cs</font>;<br/><b>assume </b><a NAME="E1:35_1_1_8"/><i><font color="Green" title="E34">A24</font></i>: 
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font>
 ;<br/><a NAME="E2:35_1_1_8"/><i><font color="Green" title="E35">A25</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c18">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c5">ccs</font>,<font color="Maroon" title="c5">ccs</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:35_1_1_8"/><i><font color="Green" title="E36">A26</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">e</font>,<font color="Maroon" title="c17">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c5">ccs</font>,<font color="Maroon" title="c5">ccs</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:35_1_1_8"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ccs</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c19">y</font> = <font color="Maroon" title="c17">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="weddwitt.html#E4:35_1_1"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/><b>thus </b><font color="Maroon" title="c17">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c18">e</font> = 
<font color="Maroon" title="c19">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E1:35_1_1_8"><i><font color="Green" title="E34">A24</font></i></a>, <a class="txt" href="weddwitt.html#E2:35_1_1_8"><i><font color="Green" title="E35">A25</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c17">x</font>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/><b>thus </b><font color="Maroon" title="c18">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c17">x</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c19">y</font>
<b>by </b><i><a class="txt" href="weddwitt.html#E1:35_1_1_8"><i><font color="Green" title="E34">A24</font></i></a>, <a class="txt" href="weddwitt.html#E3:35_1_1_8"><i><font color="Green" title="E36">A26</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c17">x</font>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c17">e</font> =  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">cs</font> ;<br/>
<a NAME="E23:35_1_1"/><i><font color="Green" title="E34">A27</font></i>: 
<font color="Maroon" title="c16">cs</font> is <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_9"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c17">e</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D2" title="GROUP_1:def.2">GROUP_1:def 2</a><br/>

<b>thus </b><a NAME="E1:35_1_1_9"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">cs</font> holds <br/> ( <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c17">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="weddwitt.html#E21:35_1_1"><i><font color="Green" title="E33">A23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:35_1_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">cs</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c17">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> )
 
<b>by </b><i><a class="txt" href="weddwitt.html#E21:35_1_1"><i><font color="Green" title="E33">A23</font></i></a></i>;<br/>
<a NAME="E25:35_1_1"/><b>then </b><i><font color="Green" title="E35">A28</font></i>: 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c16">cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E23:35_1_1"><i><font color="Green" title="E34">A27</font></i></a>, <a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c18">ccs1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">cs</font>;<br/>
<div><i><font color="Green" title="E36">A29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/><b>assume </b><a NAME="E1:35_1_1_10"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ccs</font>
 ;<br/><a NAME="E2:35_1_1_10"/><b>then </b>
 ex <font color="Olive" title="b1">x'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b1">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x</font> &amp; <font color="Olive" title="b1">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">x'</font> )
 ;<br/><b>hence </b><a NAME="E3:35_1_1_10"/>
<font color="Maroon" title="c19">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c19">x</font>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E37">A30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">a</font>, <font color="Maroon" title="c20">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c21">c</font>, <font color="Maroon" title="c22">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">cs</font>;<br/><b>assume </b><a NAME="E1:35_1_1_11"/><i><font color="Green" title="E38">A31</font></i>: 
( <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">c</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">d</font> )
 ;<br/><a NAME="E2:35_1_1_11"/><i><font color="Green" title="E39">A32</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">c</font>,<font color="Maroon" title="c22">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c5">ccs</font>,<font color="Maroon" title="c5">ccs</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>thus </b><a NAME="E3:35_1_1_11"/>
<font color="Maroon" title="c19">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">c</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c22">d</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:35_1_1_11"><i><font color="Green" title="E38">A31</font></i></a>, <a class="txt" href="weddwitt.html#E2:35_1_1_11"><i><font color="Green" title="E39">A32</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div>
<a NAME="E28:35_1_1"/><i><font color="Green" title="E38">A33</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">cs</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">d</font> holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b4">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">a</font>, <font color="Maroon" title="c20">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c21">c</font>, <font color="Maroon" title="c22">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">cs</font>;<br/>



<b>assume </b><a NAME="E1:35_1_1_12"/><i><font color="Green" title="E39">A34</font></i>: 
( <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">c</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">d</font> )
 ;<br/>

<a NAME="E2:35_1_1_12"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">c</font>,<font color="Maroon" title="c22">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c5">ccs</font>,<font color="Maroon" title="c5">ccs</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:35_1_1_12"/>
<font color="Maroon" title="c19">a</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">c</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c22">d</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:35_1_1_12"><i><font color="Green" title="E39">A34</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E29:35_1_1"/><i><font color="Green" title="E39">A35</font></i>: 
<font color="Maroon" title="c16">cs</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x</font>, <font color="Maroon" title="c20">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">cs</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a><br/>

<a NAME="E1:35_1_1_13"/>
( <font color="Maroon" title="c19">x</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="c16">cs</font> &amp; <font color="Maroon" title="c20">y</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="c16">cs</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c21">x'</font> = <font color="Maroon" title="c19">x</font>, <font color="Maroon" title="c22">y'</font> = <font color="Maroon" title="c20">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="weddwitt.html#E4:35_1_1"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon" title="c19">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c20">y</font> = 
<font color="Maroon" title="c22">y'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c21">x'</font>
<b>by </b><i><a class="txt" href="weddwitt.html#E28:35_1_1"><i><font color="Green" title="E38">A33</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c20">y</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c19">x</font>
<b>by </b><i><a class="txt" href="weddwitt.html#E28:35_1_1"><i><font color="Green" title="E38">A33</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E30:35_1_1"/><i><font color="Green" title="E40">A36</font></i>: 
<font color="Maroon" title="c16">cs</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x</font>, <font color="Maroon" title="c20">y</font>, <font color="Maroon" title="c21">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">cs</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a><br/>

<a NAME="E1:35_1_1_14"/>
( <font color="Maroon" title="c19">x</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="c16">cs</font> &amp; <font color="Maroon" title="c20">y</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="c16">cs</font> &amp; <font color="Maroon" title="c21">z</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="c16">cs</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c22">x'</font> = <font color="Maroon" title="c19">x</font>, <font color="Maroon" title="c23">y'</font> = <font color="Maroon" title="c20">y</font>, <font color="Maroon" title="c24">z'</font> = <font color="Maroon" title="c21">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="weddwitt.html#E4:35_1_1"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>
<a NAME="E3:35_1_1_14"/><i><font color="Green" title="E41">A37</font></i>: 
<font color="Maroon" title="c22">x'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c23">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c20">y</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E28:35_1_1"><i><font color="Green" title="E38">A33</font></i></a></i>;<br/>
<a NAME="E4:35_1_1_14"/><i><font color="Green" title="E42">A38</font></i>: 
<font color="Maroon" title="c23">y'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c24">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">y</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c21">z</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E28:35_1_1"><i><font color="Green" title="E38">A33</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c19">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c20">y</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c21">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">x'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c23">y'</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c24">z'</font>
<b>by </b><i><a class="txt" href="weddwitt.html#E28:35_1_1"><i><font color="Green" title="E38">A33</font></i></a>, <a class="txt" href="weddwitt.html#E3:35_1_1_14"><i><font color="Green" title="E41">A37</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c22">x'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">y'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c24">z'</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/>.= 
<font color="Maroon" title="c19">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">y</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c21">z</font></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E28:35_1_1"><i><font color="Green" title="E38">A33</font></i></a>, <a class="txt" href="weddwitt.html#E4:35_1_1_14"><i><font color="Green" title="E42">A38</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E31:35_1_1"/><i><font color="Green" title="E41">A39</font></i>: 
<font color="Maroon" title="c16">cs</font> is <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_15"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">cs</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a><br/>

<a NAME="E1:35_1_1_15"/>
<font color="Maroon" title="c19">x</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="c16">cs</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">x'</font> = <font color="Maroon" title="c19">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="weddwitt.html#E4:35_1_1"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon" title="c19">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c16">cs</font></span>)</span> = 
<font color="Maroon" title="c20">x'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E28:35_1_1"><i><font color="Green" title="E38">A33</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c19">x</font>
<b>by </b><i><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>
<a NAME="E32:35_1_1"/><i><font color="Green" title="E42">A40</font></i>: 
<font color="Maroon" title="c16">cs</font> is <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_16"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">cs</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a><br/>

<a NAME="E1:35_1_1_16"/>
<font color="Maroon" title="c19">x</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="c16">cs</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">x'</font> = <font color="Maroon" title="c19">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="weddwitt.html#E4:35_1_1"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">y'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E4:35_1_1_16"/><i><font color="Green" title="E43">A41</font></i>: 
<font color="Maroon" title="c20">x'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c21">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></i>;<br/>
<a NAME="E5:35_1_1_16"/><i><font color="Green" title="E44">A42</font></i>: 
<font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E26:35_1_1"><i><font color="Green" title="E36">A29</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> = 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>
<br/>.= 
<font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/>
<a NAME="E7:35_1_1_16"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</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="c21">y'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c21">y'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E4:35_1_1_16"><i><font color="Green" title="E43">A41</font></i></a>, <a class="ref" href="vectsp_1.html#D12" title="VECTSP_1:def.12">VECTSP_1:def 12</a></i>;<br/>
<a NAME="E8:35_1_1_16"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</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="c21">y'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">x'</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="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D11" title="VECTSP_1:def.11">VECTSP_1:def 11</a></i>;<br/>
<a NAME="E9:35_1_1_16"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">y'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">x'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">x'</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E5:35_1_1_16"><i><font color="Green" title="E44">A42</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>;<br/>
<a NAME="E10:35_1_1_16"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</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="c21">y'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">x'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">x'</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>;<br/>
<a NAME="E11:35_1_1_16"/><b>then </b>
<font color="Maroon" title="c21">y'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">x'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">x'</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>;<br/>
<a NAME="E12:35_1_1_16"/><b>then </b>
<font color="Maroon" title="c21">y'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">x'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">x'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</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="E13:35_1_1_16"/><b>then </b>
<font color="Maroon" title="c21">y'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</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="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>;<br/>
<a NAME="E14:35_1_1_16"/><b>then </b>
<font color="Maroon" title="c21">y'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>;<br/>
<a NAME="E15:35_1_1_16"/><b>then </b>
<font color="Maroon" title="c21">y'</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="c16">cs</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c22">y</font> = <font color="Maroon" title="c21">y'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">cs</font> ;<br/>
<a NAME="E17:35_1_1_16"/>
<font color="Maroon" title="c20">x'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c21">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c22">y</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E28:35_1_1"><i><font color="Green" title="E38">A33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:35_1_1_16"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">cs</font> st <font color="Maroon" title="c19">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c16">cs</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:35_1_1_16"><i><font color="Green" title="E43">A41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E33:35_1_1"/><i><font color="Green" title="E43">A43</font></i>: 
<font color="Maroon" title="c16">cs</font> is <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_17"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x</font>, <font color="Maroon" title="c20">y</font>, <font color="Maroon" title="c21">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">cs</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a><br/>

<a NAME="E1:35_1_1_17"/>
( <font color="Maroon" title="c19">x</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="c16">cs</font> &amp; <font color="Maroon" title="c20">y</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="c16">cs</font> &amp; <font color="Maroon" title="c21">z</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="c16">cs</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c22">x'</font> = <font color="Maroon" title="c19">x</font>, <font color="Maroon" title="c23">y'</font> = <font color="Maroon" title="c20">y</font>, <font color="Maroon" title="c24">z'</font> = <font color="Maroon" title="c21">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="weddwitt.html#E4:35_1_1"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>
<a NAME="E3:35_1_1_17"/><i><font color="Green" title="E44">A44</font></i>: 
<font color="Maroon" title="c22">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c23">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">y</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E27:35_1_1"><i><font color="Green" title="E37">A30</font></i></a></i>;<br/>
<a NAME="E4:35_1_1_17"/><i><font color="Green" title="E45">A45</font></i>: 
<font color="Maroon" title="c23">y'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c24">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">z</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E27:35_1_1"><i><font color="Green" title="E37">A30</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c19">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">y</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c23">y'</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c24">z'</font>
<b>by </b><i><a class="txt" href="weddwitt.html#E27:35_1_1"><i><font color="Green" title="E37">A30</font></i></a>, <a class="txt" href="weddwitt.html#E3:35_1_1_17"><i><font color="Green" title="E44">A44</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c22">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">y'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c24">z'</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c19">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">z</font></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E27:35_1_1"><i><font color="Green" title="E37">A30</font></i></a>, <a class="txt" href="weddwitt.html#E4:35_1_1_17"><i><font color="Green" title="E45">A45</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E34:35_1_1"/><i><font color="Green" title="E44">A46</font></i>: 
<font color="Maroon" title="c16">cs</font> is <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">right_unital</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_18"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">cs</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a><br/>

<a NAME="E1:35_1_1_18"/>
<font color="Maroon" title="c19">x</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="c16">cs</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">x'</font> = <font color="Maroon" title="c19">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="weddwitt.html#E4:35_1_1"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon" title="c19">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c16">cs</font></span>)</span> = 
<font color="Maroon" title="c20">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E25:35_1_1"><i><font color="Green" title="E35">A28</font></i></a>, <a class="txt" href="weddwitt.html#E27:35_1_1"><i><font color="Green" title="E37">A30</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c19">x</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E35:35_1_1"/><i><font color="Green" title="E45">A47</font></i>: 
<font color="Maroon" title="c16">cs</font> is <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">left_unital</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_19"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">cs</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D19" title="VECTSP_1:def.19">VECTSP_1:def 19</a><br/>

<a NAME="E1:35_1_1_19"/>
<font color="Maroon" title="c19">x</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="c16">cs</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">x'</font> = <font color="Maroon" title="c19">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="weddwitt.html#E4:35_1_1"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c16">cs</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c19">x</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">x'</font>
<b>by </b><i><a class="txt" href="weddwitt.html#E25:35_1_1"><i><font color="Green" title="E35">A28</font></i></a>, <a class="txt" href="weddwitt.html#E27:35_1_1"><i><font color="Green" title="E37">A30</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c19">x</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19" title="VECTSP_1:def.19">VECTSP_1:def 19</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E36:35_1_1"/><i><font color="Green" title="E46">A48</font></i>: 
<font color="Maroon" title="c16">cs</font> is <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">distributive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_20"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x</font>, <font color="Maroon" title="c20">y</font>, <font color="Maroon" title="c21">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">cs</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a><br/>

<a NAME="E1:35_1_1_20"/>
( <font color="Maroon" title="c19">x</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="c16">cs</font> &amp; <font color="Maroon" title="c20">y</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="c16">cs</font> &amp; <font color="Maroon" title="c21">z</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="c16">cs</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c22">x'</font> = <font color="Maroon" title="c19">x</font>, <font color="Maroon" title="c23">y'</font> = <font color="Maroon" title="c20">y</font>, <font color="Maroon" title="c24">z'</font> = <font color="Maroon" title="c21">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="weddwitt.html#E4:35_1_1"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>
<a NAME="E3:35_1_1_20"/><i><font color="Green" title="E47">A49</font></i>: 
<font color="Maroon" title="c20">y</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">y'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c24">z'</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E28:35_1_1"><i><font color="Green" title="E38">A33</font></i></a></i>;<br/>
<a NAME="E4:35_1_1_20"/><i><font color="Green" title="E48">A50</font></i>: 
<font color="Maroon" title="c22">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c23">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">y</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E27:35_1_1"><i><font color="Green" title="E37">A30</font></i></a></i>;<br/>
<a NAME="E5:35_1_1_20"/><i><font color="Green" title="E49">A51</font></i>: 
<font color="Maroon" title="c22">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c24">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">z</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E27:35_1_1"><i><font color="Green" title="E37">A30</font></i></a></i>;<br/>
<a NAME="E6:35_1_1_20"/><i><font color="Green" title="E50">A52</font></i>: 
<font color="Maroon" title="c23">y'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c22">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c19">x</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E27:35_1_1"><i><font color="Green" title="E37">A30</font></i></a></i>;<br/>
<a NAME="E7:35_1_1_20"/><i><font color="Green" title="E51">A53</font></i>: 
<font color="Maroon" title="c24">z'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c22">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">z</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c19">x</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E27:35_1_1"><i><font color="Green" title="E37">A30</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon" title="c19">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">y</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c21">z</font></span>)</span> = 
<font color="Maroon" title="c22">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">y'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c24">z'</font></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E27:35_1_1"><i><font color="Green" title="E37">A30</font></i></a>, <a class="txt" href="weddwitt.html#E3:35_1_1_20"><i><font color="Green" title="E47">A49</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c23">y'</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="c22">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c24">z'</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">y</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">z</font></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E28:35_1_1"><i><font color="Green" title="E38">A33</font></i></a>, <a class="txt" href="weddwitt.html#E4:35_1_1_20"><i><font color="Green" title="E48">A50</font></i></a>, <a class="txt" href="weddwitt.html#E5:35_1_1_20"><i><font color="Green" title="E49">A51</font></i></a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c20">y</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c21">z</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c19">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">y'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c24">z'</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c22">x'</font>
<b>by </b><i><a class="txt" href="weddwitt.html#E27:35_1_1"><i><font color="Green" title="E37">A30</font></i></a>, <a class="txt" href="weddwitt.html#E3:35_1_1_20"><i><font color="Green" title="E47">A49</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">y'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c22">x'</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="c24">z'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c22">x'</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c19">x</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c19">x</font></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E28:35_1_1"><i><font color="Green" title="E38">A33</font></i></a>, <a class="txt" href="weddwitt.html#E6:35_1_1_20"><i><font color="Green" title="E50">A52</font></i></a>, <a class="txt" href="weddwitt.html#E7:35_1_1_20"><i><font color="Green" title="E51">A53</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E37:35_1_1"/><i><font color="Green" title="E47">A54</font></i>: 
<font color="Maroon" title="c16">cs</font> is <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">Field-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_21"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">cs</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D20" title="VECTSP_1:def.20">VECTSP_1:def 20</a><br/>

<b>assume </b><a NAME="E1:35_1_1_21"/><i><font color="Green" title="E48">A55</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c16">cs</font>
 ;<br/>

<a NAME="E2:35_1_1_21"/>
<font color="Maroon" title="c19">x</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="c16">cs</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">x'</font> = <font color="Maroon" title="c19">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="weddwitt.html#E4:35_1_1"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">y'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E5:35_1_1_21"/><i><font color="Green" title="E49">A56</font></i>: 
<font color="Maroon" title="c20">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E12:35_1_1"><i><font color="Green" title="E31">A21</font></i></a>, <a class="txt" href="weddwitt.html#E1:35_1_1_21"><i><font color="Green" title="E48">A55</font></i></a>, <a class="ref" href="vectsp_1.html#D20" title="VECTSP_1:def.20">VECTSP_1:def 20</a></i>;<br/>
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> = 
<font color="Maroon" title="c2">s</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19" title="VECTSP_1:def.19">VECTSP_1:def 19</a></i>
<br/>.= 
<font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>
;<br/>
<a NAME="E7:35_1_1_21"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">x'</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E5:35_1_1_21"><i><font color="Green" title="E49">A56</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>;<br/>
<a NAME="E8:35_1_1_21"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E26:35_1_1"><i><font color="Green" title="E36">A29</font></i></a></i>;<br/>
<a NAME="E9:35_1_1_21"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="vectsp_2.html#K1" title="VECTSP_2:func.1">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="vectsp_2.html#K1" title="VECTSP_2:func.1">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>;<br/>
<a NAME="E10:35_1_1_21"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="vectsp_2.html#K1" title="VECTSP_2:func.1">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="vectsp_2.html#K1" title="VECTSP_2:func.1">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font>
 
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>;<br/>
<a NAME="E11:35_1_1_21"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="vectsp_2.html#K1" title="VECTSP_2:func.1">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">x'</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="vectsp_2.html#K1" title="VECTSP_2:func.1">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font>
 
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>;<br/>
<a NAME="E12:35_1_1_21"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="vectsp_2.html#K1" title="VECTSP_2:func.1">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">x'</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="vectsp_2.html#K1" title="VECTSP_2:func.1">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">x'</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font>
 
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>;<br/>
<a NAME="E13:35_1_1_21"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">x'</font> <a href="vectsp_2.html#K1" title="VECTSP_2:func.1">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">x'</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E12:35_1_1"><i><font color="Green" title="E31">A21</font></i></a>, <a class="txt" href="weddwitt.html#E1:35_1_1_21"><i><font color="Green" title="E48">A55</font></i></a>, <a class="ref" href="vectsp_2.html#T43" title="VECTSP_2:th.43">VECTSP_2:43</a></i>;<br/>
<a NAME="E14:35_1_1_21"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E12:35_1_1"><i><font color="Green" title="E31">A21</font></i></a>, <a class="txt" href="weddwitt.html#E1:35_1_1_21"><i><font color="Green" title="E48">A55</font></i></a>, <a class="ref" href="vectsp_2.html#T43" title="VECTSP_2:th.43">VECTSP_2:43</a></i>;<br/>
<a NAME="E15:35_1_1_21"/><b>then </b>
<font color="Maroon" title="c21">y'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D19" title="VECTSP_1:def.19">VECTSP_1:def 19</a></i>;<br/>
<a NAME="E16:35_1_1_21"/><b>then </b>
<font color="Maroon" title="c21">y'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">y'</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D19" title="VECTSP_1:def.19">VECTSP_1:def 19</a></i>;<br/>
<a NAME="E17:35_1_1_21"/><b>then </b>
<font color="Maroon" title="c21">y'</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="c16">cs</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c22">y</font> = <font color="Maroon" title="c21">y'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">cs</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c22">y</font>
;<br/>

<b>thus </b><a NAME="E19:35_1_1_21"/>
<font color="Maroon" title="c19">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c16">cs</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E25:35_1_1"><i><font color="Green" title="E35">A28</font></i></a>, <a class="txt" href="weddwitt.html#E27:35_1_1"><i><font color="Green" title="E37">A30</font></i></a>, <a class="txt" href="weddwitt.html#E5:35_1_1_21"><i><font color="Green" title="E49">A56</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E38:35_1_1"/>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D21" title="VECTSP_1:def.21">VECTSP_1:def 21</a></i>;<br/>
<a NAME="E39:35_1_1"/><b>then </b>
not <font color="Maroon" title="c16">cs</font> is <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">degenerated</a>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E20:35_1_1"><i><font color="Green" title="E32">A22</font></i></a>, <a class="txt" href="weddwitt.html#E25:35_1_1"><i><font color="Green" title="E35">A28</font></i></a>, <a class="ref" href="vectsp_1.html#D21" title="VECTSP_1:def.21">VECTSP_1:def 21</a></i>;<br/>
<b>hence </b><a NAME="E40:35_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">strict</a> <a href="vectsp_2.html#NM3" title="VECTSP_2:NM.3">Skew-Field</a> st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">x</font> </span>}</span>  &amp; the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">R</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> &amp; the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">R</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> &amp; the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c1">R</font> &amp; the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">unity</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="txt" href="weddwitt.html#E29:35_1_1"><i><font color="Green" title="E39">A35</font></i></a>, <a class="txt" href="weddwitt.html#E30:35_1_1"><i><font color="Green" title="E40">A36</font></i></a>, <a class="txt" href="weddwitt.html#E31:35_1_1"><i><font color="Green" title="E41">A39</font></i></a>, <a class="txt" href="weddwitt.html#E32:35_1_1"><i><font color="Green" title="E42">A40</font></i></a>, <a class="txt" href="weddwitt.html#E33:35_1_1"><i><font color="Green" title="E43">A43</font></i></a>, <a class="txt" href="weddwitt.html#E34:35_1_1"><i><font color="Green" title="E44">A46</font></i></a>, <a class="txt" href="weddwitt.html#E35:35_1_1"><i><font color="Green" title="E45">A47</font></i></a>, <a class="txt" href="weddwitt.html#E36:35_1_1"><i><font color="Green" title="E46">A48</font></i></a>, <a class="txt" href="weddwitt.html#E37:35_1_1"><i><font color="Green" title="E47">A54</font></i></a></i>;<br/>


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