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

<b>set </b><font color="Maroon">cR</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">ccs</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds  <font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> <a href="group_1.html#K1">*</a> <font color="Olive">x</font> </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E2:24_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds  <font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> <a href="group_1.html#K1">*</a> <font color="Olive">x</font> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">ccs</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds  <font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> <a href="group_1.html#K1">*</a> <font color="Olive">x</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E4:24_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">ccs</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:24_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ccs</font>
 ;<br/>

<a NAME="E2:24_1_1_2"/><b>then </b>
 ex <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds  <font color="Olive">x'</font> <a href="group_1.html#K1">*</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> <a href="group_1.html#K1">*</a> <font color="Olive">x'</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:24_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">acs</font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font> <a href="realset1.html#K1">||</a> <font color="Maroon">ccs</font>;<br/>
<b>set </b><font color="Maroon">mcs</font> = the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> <a href="realset1.html#K1">||</a> <font color="Maroon">ccs</font>;<br/>
<b>set </b><font color="Maroon">Zs</font> = the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">Us</font> =  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>;<br/>
<a NAME="E5:24_1_1"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">ccs</font>,<font color="Maroon">ccs</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:24_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">ccs</font>,<font color="Maroon">ccs</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_1_1_3"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ccs</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ccs</font> )
 <b>and </b><br/><a NAME="E4:24_1_1_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>thus </b><a NAME="E5:24_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">acs</font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font> <a href="realset1.html#K1">||</a> <font color="Maroon">ccs</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">ccs</font>,<font color="Maroon">ccs</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<a NAME="E7:24_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">acs</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">ccs</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:24_1_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">acs</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_1_1_4"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">acs</font>
 <b>and </b><br/><a NAME="E4:24_1_1_4"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">acs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:24_1_1_4"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ccs</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ccs</font> )
 <b>and </b><br/><a NAME="E7:24_1_1_4"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i>, </i>;<br/>
<a NAME="E9:24_1_1_4"/><i><font color="Green">E55</font></i>: 
 ex <font color="Olive">a'</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; ( for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds  <font color="Olive">a'</font> <a href="group_1.html#K1">*</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> <a href="group_1.html#K1">*</a> <font color="Olive">a'</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E10:24_1_1_4"/><i><font color="Green">E56</font></i>: 
 ex <font color="Olive">b'</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds  <font color="Olive">b'</font> <a href="group_1.html#K1">*</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> <a href="group_1.html#K1">*</a> <font color="Olive">b'</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E11:24_1_1_4"/><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">ccs</font>,<font color="Maroon">ccs</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E12:24_1_1_4"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">acs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D12">VECTSP_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D11">VECTSP_1:def 11</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:24_1_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ccs</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">acs</font> = <font color="Maroon">acs</font> as   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">ccs</font> <b>by </b><i><a class="ref" href="funct_2.html#T8">FUNCT_2:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">mcs</font> = the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> <a href="realset1.html#K1">||</a> <font color="Maroon">ccs</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">ccs</font>,<font color="Maroon">ccs</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<a NAME="E10:24_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">mcs</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">ccs</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:24_1_1_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">mcs</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_1_1_5"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">mcs</font>
 <b>and </b><br/><a NAME="E4:24_1_1_5"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mcs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:24_1_1_5"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ccs</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ccs</font> )
 <b>and </b><br/><a NAME="E7:24_1_1_5"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i>, </i>;<br/>
<a NAME="E9:24_1_1_5"/><i><font color="Green">E64</font></i>: 
 ex <font color="Olive">a'</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; ( for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds  <font color="Olive">a'</font> <a href="group_1.html#K1">*</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> <a href="group_1.html#K1">*</a> <font color="Olive">a'</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E10:24_1_1_5"/><i><font color="Green">E65</font></i>: 
 ex <font color="Olive">b'</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds  <font color="Olive">b'</font> <a href="group_1.html#K1">*</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> <a href="group_1.html#K1">*</a> <font color="Olive">b'</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E11:24_1_1_5"/><i><font color="Green">E66</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">ccs</font>,<font color="Maroon">ccs</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E12:24_1_1_5"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mcs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> = 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:24_1_1_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ccs</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">mcs</font> = <font color="Maroon">mcs</font> as   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">ccs</font> <b>by </b><i><a class="ref" href="funct_2.html#T8">FUNCT_2:8</a></i>;<br/>
<a NAME="E12:24_1_1"/><i><font color="Green">E68</font></i>: 
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 
;<br/>
<b>reconsider </b><font color="Maroon">Zs</font> = the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">ccs</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> = 
<font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E15:24_1_1"/><b>then </b>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ccs</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">Us</font> =  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">ccs</font> ;<br/>
<b>reconsider </b><font color="Maroon">cs</font> =  <a href="vectsp_1.html#G3">doubleLoopStr</a>(# <font color="Maroon">ccs</font>,<font color="Maroon">acs</font>,<font color="Maroon">mcs</font>,<font color="Maroon">Us</font>,<font color="Maroon">Zs</font> #) as  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a>  ;<br/>
<a NAME="E18:24_1_1"/><i><font color="Green">E69</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">cs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Zs</font>
 
;<br/>
<b>set </b><font color="Maroon">ccs1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font>;<br/>
<div><i><font color="Green">E70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>, <font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/><b>assume </b><a NAME="E1:24_1_1_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ccs</font>
 ;<br/><a NAME="E2:24_1_1_7"/><b>then </b>
 ex <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds  <font color="Olive">x'</font> <a href="group_1.html#K1">*</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> <a href="group_1.html#K1">*</a> <font color="Olive">x'</font> ) )
 ;<br/><b>hence </b><a NAME="E3:24_1_1_7"/>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font>, <font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font>;<br/><b>assume </b><a NAME="E1:24_1_1_8"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 ;<br/><a NAME="E2:24_1_1_8"/><i><font color="Green">E74</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">ccs</font>,<font color="Maroon">ccs</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>thus </b><a NAME="E3:24_1_1_8"/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <font color="Maroon">d</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div>
<a NAME="E21:24_1_1"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><br/> for <font color="Olive">c</font>, <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">d</font> holds <br/><font color="Olive">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">d</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E22:24_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">cs</font> is <a href="rlvect_1.html#V3">Abelian</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_9"><b>proof </b></a><div class="add">

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

<a NAME="E1:24_1_1_9"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> = 
<font color="Maroon">y'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E23:24_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">cs</font> is <a href="rlvect_1.html#V4">add-associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_10"><b>proof </b></a><div class="add">

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

<a NAME="E1:24_1_1_10"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E3:24_1_1_10"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">x'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:24_1_1_10"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">y'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y'</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z'</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">x'</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z'</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E24:24_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">cs</font> is <a href="rlvect_1.html#V5">right_zeroed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_11"><b>proof </b></a><div class="add">

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

<a NAME="E1:24_1_1_11"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">cs</font></span>)</span> = 
<font color="Maroon">x'</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E25:24_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">cs</font> is <a href="rlvect_1.html#V6">right_complementable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_12"><b>proof </b></a><div class="add">

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

<a NAME="E1:24_1_1_12"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y'</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E4:24_1_1_12"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">x'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/><a NAME="E1:24_1_1_12_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
;<br/><a NAME="E3:24_1_1_12_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="vectsp_1.html#D12">VECTSP_1:def 12</a></i>;<br/><a NAME="E4:24_1_1_12_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D11">VECTSP_1:def 11</a></i>;<br/><a NAME="E5:24_1_1_12_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="weddwitt.html#T1" target="_self">Th1</a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/><a NAME="E6:24_1_1_12_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>;<br/><a NAME="E7:24_1_1_12_1"/><b>then </b>
<font color="Maroon">y'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/><a NAME="E8:24_1_1_12_1"/><b>then </b>
<font color="Maroon">y'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/><a NAME="E9:24_1_1_12_1"/><b>then </b>
<font color="Maroon">y'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>;<br/><b>hence </b><a NAME="E10:24_1_1_12_1"/>
<font color="Maroon">y'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:24_1_1_12"/><b>then </b>
<font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y'</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font> ;<br/>
<a NAME="E8:24_1_1_12"/>
<font color="Maroon">x'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:24_1_1_12"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font> st <font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">cs</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E26:24_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">cs</font> is <a href="group_1.html#V4">associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_13"><b>proof </b></a><div class="add">

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

<a NAME="E1:24_1_1_13"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E3:24_1_1_13"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:24_1_1_13"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">y'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z'</font>
<b>by </b><i>, <a class="ref" href="weddwitt.html#T1" target="_self">Th1</a></i>
<br/>.= 
<font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z'</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E27:24_1_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">cs</font> is <a href="group_1.html#V7">commutative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D16">GROUP_1:def 16</a><br/>

<a NAME="E1:24_1_1_14"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> = 
<font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">y'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E103</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_15"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">cs</font>, <font color="Maroon">e</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">cs</font>;<br/><b>assume </b><a NAME="E1:24_1_1_15"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 ;<br/><a NAME="E2:24_1_1_15"/><i><font color="Green">E105</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">e</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">ccs</font>,<font color="Maroon">ccs</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:24_1_1_15"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ccs</font>
 ;<br/><b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/><b>thus </b><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> = 
<font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i>, <a class="txt" href="weddwitt.html#E2:24_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/><b>hence </b><a NAME="E6:24_1_1_15"/>
<font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="weddwitt.html#T1" target="_self">Th1</a>, <a class="ref" href="group_1.html#D16">GROUP_1:def 16</a></i>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">e</font> =  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">cs</font> ;<br/>
<a NAME="E30:24_1_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">cs</font> is <a href="group_1.html#V2">unital</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_16"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:24_1_1_16"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font> holds <br/> ( <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E31:24_1_1"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">cs</font> holds <br/> ( <font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E32:24_1_1"/><b>then </b><i><font color="Green">E107</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">cs</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>;<br/>
<a NAME="E33:24_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">cs</font> is <a href="vectsp_1.html#V8">left_unital</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_17"><b>proof </b></a><div class="add">

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

<a NAME="E1:24_1_1_17"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">cs</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font>
<b>by </b><i><a class="txt" href="weddwitt.html#E2:24_1_1"><i><font color="Green">E35</font></i></a>, </i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E34:24_1_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">cs</font> is <a href="vectsp_1.html#V7">distributive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_18"><b>proof </b></a><div class="add">

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

<a NAME="E1:24_1_1_18"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E3:24_1_1_18"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:24_1_1_18"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:24_1_1_18"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:24_1_1_18"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">y'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:24_1_1_18"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">z'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z'</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z'</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, <a class="ref" href="weddwitt.html#T2" target="_self">Th2</a>, </i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z'</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E35:24_1_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">cs</font> is <a href="vectsp_1.html#V9">Field-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_19"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:24_1_1_19"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">cs</font>
 ;<br/>

<a NAME="E2:24_1_1_19"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y'</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E5:24_1_1_19"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="weddwitt.html#T2" target="_self">Th2</a>, <a class="ref" href="vectsp_1.html#D20">VECTSP_1:def 20</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_19_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/><span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> = 
<font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
;<br/><a NAME="E2:24_1_1_19_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font>
 <b>by </b><i>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/><a NAME="E3:24_1_1_19_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font>
 <b>by </b><i/>;<br/><a NAME="E4:24_1_1_19_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="vectsp_2.html#K1">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="vectsp_2.html#K1">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/><a NAME="E5:24_1_1_19_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="vectsp_2.html#K1">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="vectsp_2.html#K1">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font>
 <b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/><a NAME="E6:24_1_1_19_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x'</font> <a href="vectsp_2.html#K1">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="vectsp_2.html#K1">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font>
 <b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/><a NAME="E7:24_1_1_19_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x'</font> <a href="vectsp_2.html#K1">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x'</font> <a href="vectsp_2.html#K1">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font>
 <b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/><a NAME="E8:24_1_1_19_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x'</font> <a href="vectsp_2.html#K1">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font>
 <b>by </b><i>, <a class="ref" href="weddwitt.html#T2" target="_self">Th2</a>, <a class="ref" href="vectsp_2.html#T43">VECTSP_2:43</a></i>;<br/><a NAME="E9:24_1_1_19_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font>
 <b>by </b><i>, <a class="ref" href="weddwitt.html#T2" target="_self">Th2</a>, <a class="ref" href="vectsp_2.html#T43">VECTSP_2:43</a></i>;<br/><a NAME="E10:24_1_1_19_1"/><b>then </b>
<font color="Maroon">y'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>;<br/><b>hence </b><a NAME="E11:24_1_1_19_1"/>
<font color="Maroon">y'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:24_1_1_19"/><b>then </b>
<font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y'</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">cs</font> ;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E9:24_1_1_19"/>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">cs</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E2:24_1_1"><i><font color="Green">E35</font></i></a>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E36:24_1_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/>
<a NAME="E37:24_1_1"/><b>then </b>
not <font color="Maroon">cs</font> is <a href="vectsp_1.html#V10">degenerated</a>
 
<b>by </b><i>, <a class="txt" href="weddwitt.html#E2:24_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/>
<b>hence </b><a NAME="E38:24_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being <a href="vectsp_1.html#V3">strict</a> <a href="vectsp_1.html#NM2">Field</a> st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds  <font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> <a href="group_1.html#K1">*</a> <font color="Olive">x</font> </span>}</span>  &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> &amp; the <a href="group_1.html#U1">mult</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> &amp; the <a href="struct_0.html#U2">Zero</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font> &amp; the <a href="vectsp_1.html#U1">unity</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> )
 <b>by </b><i>, , , , , <a class="txt" href="weddwitt.html#E4:24_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="weddwitt.html#E5:24_1_1"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="weddwitt.html#T1" target="_self">Th1</a></i>;<br/>


</div>
