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<div class="add">

<b>let </b><font color="Maroon">R</font> be   <a href="gcd_1.html#NM3">gcdDomain</a>;<br/><b>let </b><font color="Maroon">Amp</font> be    <a href="gcd_1.html#M2" target="_self">AmpleSet</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">r1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">r2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">s1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">s2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>





<b>assume </b><a NAME="E1:81"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">Amp</font> is <a href="gcd_1.html#V2" target="_self">multiplicative</a> &amp; <font color="Maroon">r1</font>,<font color="Maroon">r2</font> <a href="gcd_1.html#R8" target="_self">are_normalized_wrt</a> <font color="Maroon">Amp</font> &amp; <font color="Maroon">s1</font>,<font color="Maroon">s2</font> <a href="gcd_1.html#R8" target="_self">are_normalized_wrt</a> <font color="Maroon">Amp</font> )
 ;<br/>

<a NAME="E2:81"/><b>then </b><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> &amp; <font color="Maroon">s2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Amp</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Amp</font> )
 
<b>by </b><i><a class="ref" href="gcd_1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E3:81"/><b>then </b><i><font color="Green">E25</font></i>: 
( <font color="Maroon">r1</font>,<font color="Maroon">r2</font> <a href="gcd_1.html#R7" target="_self">are_co-prime</a>  &amp; <font color="Maroon">s1</font>,<font color="Maroon">s2</font> <a href="gcd_1.html#R7" target="_self">are_co-prime</a>  &amp; <font color="Maroon">r2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp; <font color="Maroon">s2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a>  <a href="gcd_1.html#K4" target="_self">NF</a> <font color="Maroon">r2</font>,<font color="Maroon">Amp</font> &amp; <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a>  <a href="gcd_1.html#K4" target="_self">NF</a> <font color="Maroon">s2</font>,<font color="Maroon">Amp</font> )
 
<b>by </b><i><a class="ref" href="gcd_1.html#T18" target="_self">Th18</a>, , , <a class="ref" href="gcd_1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E4:81"/><b>then </b><i><font color="Green">E26</font></i>: 
(  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</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/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_1_1"/>
( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> or <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> or ( <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> &amp; <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> ) or ( <font color="Maroon">s2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> ) or ( <font color="Maroon">r2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp; <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> ) or ( not <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; not <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; (  not <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> or  not <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> ) &amp; (  not <font color="Maroon">s2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> or  not <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> ) &amp; (  not <font color="Maroon">r2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> or  not <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_1"><b>case </b></a><a NAME="E1:81_1_1_1"/>
( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> or <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 ;<br/><div class="add"><a NAME="E2:81_1_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="gcd_1.html#K9" target="_self">mult2</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<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="ref" href="gcd_1.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E3:81_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K8" target="_self">mult1</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>,<span class="p1">(<span class="default"><a href="gcd_1.html#K9" target="_self">mult2</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>,<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="E4:81_1_1_1"/><i><font color="Green">E61</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Amp</font>
 <b>by </b><i/>;<br/><a NAME="E5:81_1_1_1"/>
 <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/><b>hence </b><a NAME="E6:81_1_1_1"/>
 <a href="gcd_1.html#K8" target="_self">mult1</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font>, <a href="gcd_1.html#K9" target="_self">mult2</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R8" target="_self">are_normalized_wrt</a> <font color="Maroon">Amp</font>
 <b>by </b><i>, <a class="ref" href="gcd_1.html#T19" target="_self">Th19</a>, <a class="ref" href="gcd_1.html#D5" target="_self">Def5</a>, <a class="ref" href="gcd_1.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2"><b>case </b></a><a NAME="E1:81_1_1_2"/>
( <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> &amp; <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> )
 ;<br/><div class="add"><a NAME="E2:81_1_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="gcd_1.html#K9" target="_self">mult2</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<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#E2:81_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="gcd_1.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E3:81_1_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K8" target="_self">mult1</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>,<span class="p1">(<span class="default"><a href="gcd_1.html#K9" target="_self">mult2</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>,<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="E4:81_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#R2">in</a> <font color="Maroon">Amp</font>
 <b>by </b><i/>;<br/><a NAME="E5:81_1_1_2"/>
 <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/><b>hence </b><a NAME="E6:81_1_1_2"/>
 <a href="gcd_1.html#K8" target="_self">mult1</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font>, <a href="gcd_1.html#K9" target="_self">mult2</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R8" target="_self">are_normalized_wrt</a> <font color="Maroon">Amp</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E4:81_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="gcd_1.html#T19" target="_self">Th19</a>, <a class="ref" href="gcd_1.html#D5" target="_self">Def5</a>, <a class="ref" href="gcd_1.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_3"><b>case </b></a><a NAME="E1:81_1_1_3"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">s2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> )
 ;<br/><div class="add"><a NAME="E2:81_1_1_3"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="gcd_1.html#K8" target="_self">mult1</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</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">s1</font></span>)</span> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>
 <b>by </b><i><a class="ref" href="gcd_1.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E3:81_1_1_3"/><i><font color="Green">E85</font></i>: 
 <a href="gcd_1.html#K9" target="_self">mult2</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="gcd_1.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E4:81_1_1_3"/><i><font color="Green">E86</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">s1</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>
 <b>by </b><i>, </i>;<br/><a NAME="E5:81_1_1_3"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:81_1_1_3_1"/>
 <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/>
<b>hence </b><a NAME="E2:81_1_1_3_1"/>
<font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E6:81_1_1_3"/><i><font color="Green">E88</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E7:81_1_1_3"/><b>then </b>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s1</font>
 <b>by </b><i/>;<br/><b>then </b><i><font color="Green">E89</font></i>: <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s1</font></span>)</span> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <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">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">s1</font>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <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">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <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">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/><a NAME="E9:81_1_1_3"/>
<font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <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">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <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">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>;<br/><a NAME="E10:81_1_1_3"/><b>then </b><i><font color="Green">E90</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K8" target="_self">mult1</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>,<span class="p1">(<span class="default"><a href="gcd_1.html#K9" target="_self">mult2</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>,<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="ref" href="gcd_1.html#T19" target="_self">Th19</a>, <a class="ref" href="gcd_1.html#D5" target="_self">Def5</a>, , <a class="ref" href="gcd_1.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E11:81_1_1_3"/><i><font color="Green">E92</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">s2</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E12:81_1_1_3"/><i><font color="Green">E93</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</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>, </i>;<br/><a NAME="E13:81_1_1_3"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span> <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>, , </i>;<br/><b>reconsider </b><font color="Maroon">zz</font> =  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font> <b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/><b>reconsider </b><font color="Maroon">s2</font> = <font color="Maroon">s2</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font> <b>by </b><i><a class="ref" href="gcd_1.html#T18" target="_self">Th18</a>, <a class="ref" href="gcd_1.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E16:81_1_1_3"/>
<font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">zz</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Amp</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T18" target="_self">Th18</a>, , <a class="ref" href="gcd_1.html#T20" target="_self">Th20</a>, </i>;<br/><b>hence </b><a NAME="E17:81_1_1_3"/>
 <a href="gcd_1.html#K8" target="_self">mult1</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font>, <a href="gcd_1.html#K9" target="_self">mult2</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R8" target="_self">are_normalized_wrt</a> <font color="Maroon">Amp</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D5" target="_self">Def5</a>, , , <a class="ref" href="gcd_1.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_4"><b>case </b></a><a NAME="E1:81_1_1_4"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">r2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp; <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> )
 ;<br/><div class="add"><a NAME="E2:81_1_1_4"/><b>then </b><i><font color="Green">E96</font></i>: 
 <a href="gcd_1.html#K8" target="_self">mult1</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</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">s1</font></span>)</span> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font></span>)</span>
 <b>by </b><i><a class="ref" href="gcd_1.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E3:81_1_1_4"/><i><font color="Green">E97</font></i>: 
 <a href="gcd_1.html#K9" target="_self">mult2</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="gcd_1.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E4:81_1_1_4"/><i><font color="Green">E99</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r1</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>, </i>;<br/><a NAME="E5:81_1_1_4"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:81_1_1_4_1"/>
 <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/>
<b>hence </b><a NAME="E2:81_1_1_4_1"/>
<font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E6:81_1_1_4"/><i><font color="Green">E101</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E7:81_1_1_4"/><b>then </b>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s1</font>
 <b>by </b><i/>;<br/><b>then </b><i><font color="Green">E102</font></i>: <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s1</font></span>)</span> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font></span>)</span> = 
<font color="Maroon">r1</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <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">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <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">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <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">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/><a NAME="E9:81_1_1_4"/>
<font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <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">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <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">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>;<br/><a NAME="E10:81_1_1_4"/><b>then </b><i><font color="Green">E103</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K8" target="_self">mult1</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>,<span class="p1">(<span class="default"><a href="gcd_1.html#K9" target="_self">mult2</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>,<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="ref" href="gcd_1.html#T19" target="_self">Th19</a>, <a class="ref" href="gcd_1.html#D5" target="_self">Def5</a>, , <a class="ref" href="gcd_1.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E11:81_1_1_4"/><i><font color="Green">E104</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">r2</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E12:81_1_1_4"/><i><font color="Green">E105</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</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>, </i>;<br/><a NAME="E13:81_1_1_4"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font></span>)</span> <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>, , </i>;<br/><b>reconsider </b><font color="Maroon">zz</font> =  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font> <b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/><b>reconsider </b><font color="Maroon">r2</font> = <font color="Maroon">r2</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font> <b>by </b><i><a class="ref" href="gcd_1.html#T18" target="_self">Th18</a>, <a class="ref" href="gcd_1.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E16:81_1_1_4"/>
<font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">zz</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Amp</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T18" target="_self">Th18</a>, , <a class="ref" href="gcd_1.html#T20" target="_self">Th20</a>, </i>;<br/><b>hence </b><a NAME="E17:81_1_1_4"/>
 <a href="gcd_1.html#K8" target="_self">mult1</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font>, <a href="gcd_1.html#K9" target="_self">mult2</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R8" target="_self">are_normalized_wrt</a> <font color="Maroon">Amp</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D5" target="_self">Def5</a>, , , <a class="ref" href="gcd_1.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_5"><b>case </b></a><a NAME="E1:81_1_1_5"/><i><font color="Green">E161</font></i>: 
( not <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; not <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; (  not <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> or  not <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> ) &amp; (  not <font color="Maroon">s2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> or  not <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> ) &amp; (  not <font color="Maroon">r2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> or  not <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> ) )
 ;<br/><div class="add"><a NAME="E2:81_1_1_5"/><b>then </b><i><font color="Green">E162</font></i>: 
 <a href="gcd_1.html#K8" target="_self">mult1</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <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">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <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">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="gcd_1.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E3:81_1_1_5"/><i><font color="Green">E163</font></i>: 
 <a href="gcd_1.html#K9" target="_self">mult2</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <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">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <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">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="gcd_1.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E4:81_1_1_5"/><b>then </b><i><font color="Green">E164</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K8" target="_self">mult1</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>,<span class="p1">(<span class="default"><a href="gcd_1.html#K9" target="_self">mult2</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span>,<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="ref" href="gcd_1.html#T19" target="_self">Th19</a>, <a class="ref" href="gcd_1.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E5:81_1_1_5"/><i><font color="Green">E165</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">s2</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E6:81_1_1_5"/><i><font color="Green">E166</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</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>, </i>;<br/><a NAME="E7:81_1_1_5"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font></span>)</span> <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>, , </i>;<br/><b>reconsider </b><font color="Maroon">z1</font> =  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">r1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font> <b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/><b>reconsider </b><font color="Maroon">s2</font> = <font color="Maroon">s2</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font> <b>by </b><i><a class="ref" href="gcd_1.html#T18" target="_self">Th18</a>, <a class="ref" href="gcd_1.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E10:81_1_1_5"/><i><font color="Green">E168</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">r2</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E11:81_1_1_5"/><i><font color="Green">E169</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</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>, </i>;<br/><a NAME="E12:81_1_1_5"/><b>then </b><i><font color="Green">E170</font></i>: 
<font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font></span>)</span> <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>, , </i>;<br/><b>reconsider </b><font color="Maroon">z2</font> =  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">s1</font>,<font color="Maroon">r2</font>,<font color="Maroon">Amp</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font> <b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/><b>reconsider </b><font color="Maroon">r2</font> = <font color="Maroon">r2</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font> <b>by </b><i><a class="ref" href="gcd_1.html#T18" target="_self">Th18</a>, <a class="ref" href="gcd_1.html#T12" target="_self">Th12</a></i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">s2</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">z1</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font> <b>by </b><i><a class="ref" href="gcd_1.html#T18" target="_self">Th18</a>, , , </i>;<br/><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">r2</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">z2</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font> <b>by </b><i><a class="ref" href="gcd_1.html#T18" target="_self">Th18</a>, , , </i>;<br/><a NAME="E17:81_1_1_5"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">v</font> <a href="group_1.html#K10">*</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Amp</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T18" target="_self">Th18</a>, </i>;<br/><a NAME="E18:81_1_1_5"/>
<font color="Maroon">v</font> <a href="group_1.html#K10">*</a> <font color="Maroon">u</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="gcd_1.html#T20" target="_self">Th20</a>, <a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a></i>;<br/><b>hence </b><a NAME="E19:81_1_1_5"/>
 <a href="gcd_1.html#K8" target="_self">mult1</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font>, <a href="gcd_1.html#K9" target="_self">mult2</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R8" target="_self">are_normalized_wrt</a> <font color="Maroon">Amp</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D5" target="_self">Def5</a>, , , <a class="ref" href="gcd_1.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:81"/>
 <a href="gcd_1.html#K8" target="_self">mult1</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font>, <a href="gcd_1.html#K9" target="_self">mult2</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R8" target="_self">are_normalized_wrt</a> <font color="Maroon">Amp</font>
 ;<br/>


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