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

<a NAME="E1:75_1_1"/><i><font color="Green">E21</font></i>: 
(  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> &amp;  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</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="gcd_1.html#E3:75_1_1"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E2:75_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp;  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> implies for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> iff <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s1</font></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:75_1_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp;  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> )
 ;<br/>

<b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

<a NAME="E2:75_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">r2</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="gcd_1.html#T14" target="_self">Th14</a>, <a class="ref" href="gcd_1.html#T17" target="_self">Th17</a>, <a class="ref" href="gcd_1.html#T2" target="_self">Th2</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s1</font></span>)</span> = 
<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">r2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s1</font></span>)</span>
<b>by </b><i><a class="ref" href="gcd_1.html#T17" target="_self">Th17</a>, <a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
<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#K10">*</a> <font color="Maroon">s1</font>
<b>by </b><i><a class="ref" href="gcd_1.html#T18" target="_self">Th18</a>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<font color="Maroon">s1</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="E4:75_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> iff <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s1</font></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:75_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> implies for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> iff <font color="Olive">z</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:75_1_1_2"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 ;<br/>

<b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

<a NAME="E2:75_1_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">r2</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="gcd_1.html#T14" target="_self">Th14</a>, <a class="ref" href="gcd_1.html#T18" target="_self">Th18</a>, <a class="ref" href="gcd_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E3:75_1_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:75_1_1_2"/><i><font color="Green">E64</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</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 href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> = 
<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">s1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="gcd_1.html#T18" target="_self">Th18</a>, <a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
<br/>.= 
<font color="Maroon">s1</font> <a href="group_1.html#K10">*</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="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gcd_1.html#E1:75_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<font color="Maroon">s1</font> <a href="group_1.html#K10">*</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="gcd_1.html#E2:75_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="gcd_1.html#E3:75_1_1_2"><i><font color="Green">E63</font></i></a>, </i>
<br/>.= 
<font color="Maroon">s1</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
;<br/>
<b>hence </b><a NAME="E6:75_1_1_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> iff <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:75_1_1"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp;  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> implies for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> iff <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s1</font></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:75_1_1_3"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp;  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> )
 ;<br/>

<b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

<a NAME="E2:75_1_1_3"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">s2</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="gcd_1.html#T14" target="_self">Th14</a>, , <a class="ref" href="gcd_1.html#T2" target="_self">Th2</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s2</font></span>)</span> <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>, <a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
<br/>.= 
<font color="Maroon">r1</font> <a href="group_1.html#K10">*</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="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<font color="Maroon">r1</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
;<br/>
<b>hence </b><a NAME="E4:75_1_1_3"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> iff <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s1</font></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:75_1_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> implies for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> iff <font color="Olive">z</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:75_1_1_4"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 ;<br/>

<b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

<a NAME="E2:75_1_1_4"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">s2</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="gcd_1.html#T14" target="_self">Th14</a>, , <a class="ref" href="gcd_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E3:75_1_1_4"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:75_1_1_4"/><i><font color="Green">E90</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</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 href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> = 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <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>, <a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
<br/>.= 
<font color="Maroon">r1</font> <a href="group_1.html#K10">*</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="gcd_1.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<font color="Maroon">r1</font> <a href="group_1.html#K10">*</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="gcd_1.html#T19" target="_self">Th19</a>, </i>
<br/>.= 
<font color="Maroon">r1</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
;<br/>
<b>hence </b><a NAME="E6:75_1_1_4"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> iff <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:75_1_1"/>
(  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> implies for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s1</font></span>)</span> iff <font color="Olive">z</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:75_1_1_5"/><i><font color="Green">E92</font></i>: 
(  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 ;<br/>

<b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> = 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r2</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E3:75_1_1_5"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s1</font></span>)</span> iff <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:75_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds <br/> ( ( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp; <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> implies ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> ) ) &amp; ( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp;  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> implies ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s1</font></span>)</span> ) ) &amp; ( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> implies ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> ) ) &amp; ( <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp;  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> implies ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s1</font></span>)</span> ) ) &amp; ( <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> implies ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> ) ) &amp; (  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r2</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> implies ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s1</font></span>)</span> iff <font color="Olive">b<sub>1</sub></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="gcd_1.html#T15" target="_self">Th15</a>, <a class="ref" href="gcd_1.html#T17" target="_self">Th17</a>, <a class="ref" href="gcd_1.html#T18" target="_self">Th18</a>, </i>;<br/>


</div>
