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<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="gcd_1.html#NM3" title="GCD_1:NM.3">gcdDomain</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Amp</font> being   <a href="gcd_1.html#M2" title="GCD_1:mode.2">AmpleSet</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</font>, <font color="Olive" title="b4">s1</font>, <font color="Olive" title="b5">s2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">Amp</font> is <a href="gcd_1.html#V2" title="GCD_1:attr.2">multiplicative</a> &amp; <font color="Olive" title="b2">r1</font>,<font color="Olive" title="b3">r2</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Olive" title="b1">Amp</font> &amp; <font color="Olive" title="b4">s1</font>,<font color="Olive" title="b5">s2</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Olive" title="b1">Amp</font> holds <br/> <a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Olive" title="b2">r1</font>,<font color="Olive" title="b3">r2</font>,<font color="Olive" title="b4">s1</font>,<font color="Olive" title="b5">s2</font>,<font color="Olive" title="b1">Amp</font>, <a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Olive" title="b2">r1</font>,<font color="Olive" title="b3">r2</font>,<font color="Olive" title="b4">s1</font>,<font color="Olive" title="b5">s2</font>,<font color="Olive" title="b1">Amp</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Olive" title="b1">Amp</font></span><br/><b>let </b><font color="Maroon" title="c2">Amp</font> be    <a href="gcd_1.html#M2" title="GCD_1:mode.2">AmpleSet</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font>, <font color="Olive" title="b3">s1</font>, <font color="Olive" title="b4">s2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Maroon" title="c2">Amp</font> is <a href="gcd_1.html#V2" title="GCD_1:attr.2">multiplicative</a> &amp; <font color="Olive" title="b1">r1</font>,<font color="Olive" title="b2">r2</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Maroon" title="c2">Amp</font> &amp; <font color="Olive" title="b3">s1</font>,<font color="Olive" title="b4">s2</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Maroon" title="c2">Amp</font> holds <br/> <a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Olive" title="b1">r1</font>,<font color="Olive" title="b2">r2</font>,<font color="Olive" title="b3">s1</font>,<font color="Olive" title="b4">s2</font>,<font color="Maroon" title="c2">Amp</font>, <a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Olive" title="b1">r1</font>,<font color="Olive" title="b2">r2</font>,<font color="Olive" title="b3">s1</font>,<font color="Olive" title="b4">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Maroon" title="c2">Amp</font></span><br/><b>let </b><font color="Maroon" title="c3">r1</font>, <font color="Maroon" title="c4">r2</font>, <font color="Maroon" title="c5">s1</font>, <font color="Maroon" title="c6">s2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">Amp</font> is <a href="gcd_1.html#V2" title="GCD_1:attr.2">multiplicative</a> &amp; <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Maroon" title="c2">Amp</font> &amp; <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Maroon" title="c2">Amp</font> implies  <a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font>, <a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Maroon" title="c2">Amp</font> )</span><br/>





<b>assume </b><a NAME="E1:81"/><i><font color="Green" title="E60">A1</font></i>: 
( <font color="Maroon" title="c2">Amp</font> is <a href="gcd_1.html#V2" title="GCD_1:attr.2">multiplicative</a> &amp; <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Maroon" title="c2">Amp</font> &amp; <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Maroon" title="c2">Amp</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font>, <a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Maroon" title="c2">Amp</font></span><br/>

<a NAME="E2:81"/><b>then </b><i><font color="Green" title="E61">A2</font></i>: 
(  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> &amp;  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Amp</font> &amp; <font color="Maroon" title="c4">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Amp</font> )
 
<b>by </b><i><a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></i>;<br/>
<a NAME="E3:81"/><b>then </b><i><font color="Green" title="E62">A3</font></i>: 
( <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#R7" title="GCD_1:pred.7">are_co-prime</a>  &amp; <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#R7" title="GCD_1:pred.7">are_co-prime</a>  &amp; <font color="Maroon" title="c4">r2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c6">s2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c4">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gcd_1.html#K4" title="GCD_1:func.4">NF</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font> &amp; <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gcd_1.html#K4" title="GCD_1:func.4">NF</a> <font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> )
 
<b>by </b><i><a class="txt" href="gcd_1.html#E1:81"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="gcd_1.html#D9" target="_self" title="GCD_1:def.9">Def9</a>, <a class="ref" href="gcd_1.html#D14" target="_self" title="GCD_1:def.14">Def14</a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></i>;<br/>
<a NAME="E4:81"/><b>then </b><i><font color="Green" title="E63">A4</font></i>: 
(  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> &amp;  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> )
 
<b>by </b><i><a class="ref" href="gcd_1.html#T34" target="_self" title="GCD_1:th.34">Th34</a></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" title="c3">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> or <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> or ( <font color="Maroon" title="c4">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> ) or ( <font color="Maroon" title="c6">s2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c4">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> ) or ( <font color="Maroon" title="c4">r2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> ) or ( not <font color="Maroon" title="c3">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> &amp; not <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> &amp; (  not <font color="Maroon" title="c4">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> or  not <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> ) &amp; (  not <font color="Maroon" title="c6">s2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> or  not <font color="Maroon" title="c4">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> ) &amp; (  not <font color="Maroon" title="c4">r2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> or  not <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">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" title="c3">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> or <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font>, <a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Maroon" title="c2">Amp</font></span><br/><div class="add"><a NAME="E2:81_1_1_1"/><b>then </b><i><font color="Green" title="E64">A5</font></i>: 
 <a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:81"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="gcd_1.html#D19" target="_self" title="GCD_1:def.19">Def19</a></i>;<br/><a NAME="E3:81_1_1_1"/><b>then </b><i><font color="Green" title="E65">A6</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>,<span class="p1">(<span class="default"><a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T33" target="_self" title="GCD_1:th.33">Th33</a></i>;<br/><a NAME="E4:81_1_1_1"/><i><font color="Green" title="E66">A7</font></i>: 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Amp</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T22" target="_self" title="GCD_1:th.22">Th22</a></i>;<br/><a NAME="E5:81_1_1_1"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 ;<br/><b>hence </b><a NAME="E6:81_1_1_1"/>
 <a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font>, <a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Maroon" title="c2">Amp</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:81_1_1_1"><i><font color="Green" title="E64">A5</font></i></a>, <a class="txt" href="gcd_1.html#E3:81_1_1_1"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="gcd_1.html#E4:81_1_1_1"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c4">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font>, <a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Maroon" title="c2">Amp</font></span><br/><div class="add"><a NAME="E2:81_1_1_2"/><b>then </b><i><font color="Green" title="E64">A8</font></i>: 
 <a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:81"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="gcd_1.html#D19" target="_self" title="GCD_1:def.19">Def19</a></i>;<br/><a NAME="E3:81_1_1_2"/><b>then </b><i><font color="Green" title="E65">A9</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>,<span class="p1">(<span class="default"><a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T33" target="_self" title="GCD_1:th.33">Th33</a></i>;<br/><a NAME="E4:81_1_1_2"/><i><font color="Green" title="E66">A10</font></i>: 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Amp</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T22" target="_self" title="GCD_1:th.22">Th22</a></i>;<br/><a NAME="E5:81_1_1_2"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 ;<br/><b>hence </b><a NAME="E6:81_1_1_2"/>
 <a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font>, <a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Maroon" title="c2">Amp</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:81_1_1_2"><i><font color="Green" title="E64">A8</font></i></a>, <a class="txt" href="gcd_1.html#E3:81_1_1_2"><i><font color="Green" title="E65">A9</font></i></a>, <a class="txt" href="gcd_1.html#E4:81_1_1_2"><i><font color="Green" title="E66">A10</font></i></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="E64">A11</font></i>: 
( <font color="Maroon" title="c6">s2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c4">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font>, <a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Maroon" title="c2">Amp</font></span><br/><div class="add"><a NAME="E2:81_1_1_3"/><b>then </b><i><font color="Green" title="E65">A12</font></i>: 
 <a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>
 <b>by </b><i><a class="ref" href="gcd_1.html#D18" target="_self" title="GCD_1:def.18">Def18</a></i>;<br/><a NAME="E3:81_1_1_3"/><i><font color="Green" title="E66">A13</font></i>: 
 <a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:81"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="gcd_1.html#E1:81_1_1_3"><i><font color="Green" title="E64">A11</font></i></a>, <a class="ref" href="gcd_1.html#D19" target="_self" title="GCD_1:def.19">Def19</a></i>;<br/><a NAME="E4:81_1_1_3"/><i><font color="Green" title="E67">A14</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:81_1_1_3"><i><font color="Green" title="E64">A11</font></i></a>, <a class="ref" href="gcd_1.html#T33" target="_self" title="GCD_1:th.33">Th33</a></i>;<br/><a NAME="E5:81_1_1_3"/><b>then </b><i><font color="Green" title="E68">A15</font></i>: 
<font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:81_1_1_3"><i><font color="Green" title="E64">A11</font></i></a>, <a class="ref" href="gcd_1.html#T9" target="_self" title="GCD_1:th.9">Th9</a></i>;<br/><a NAME="E6:81_1_1_3"/><i><font color="Green" title="E69">A16</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">r1</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></i>;<br/><a NAME="E7:81_1_1_3"/><b>then </b>
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">r1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">s1</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T7" target="_self" title="GCD_1:th.7">Th7</a></i>;<br/><b>then </b><i><font color="Green" title="E70">A17</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">s1</font>
<b>by </b><i><a class="txt" href="gcd_1.html#E4:81"><i><font color="Green" title="E63">A4</font></i></a>, <a class="txt" href="gcd_1.html#E6:81_1_1_3"><i><font color="Green" title="E69">A16</font></i></a>, <a class="ref" href="gcd_1.html#T11" target="_self" title="GCD_1:th.11">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gcd_1.html#E4:81_1_1_3"><i><font color="Green" title="E67">A14</font></i></a>, <a class="ref" href="gcd_1.html#T10" target="_self" title="GCD_1:th.10">Th10</a></i>
;<br/><a NAME="E9:81_1_1_3"/>
<font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="gcd_1.html#E5:81_1_1_3"><i><font color="Green" title="E68">A15</font></i></a>, <a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>;<br/><a NAME="E10:81_1_1_3"/><b>then </b><i><font color="Green" title="E71">A18</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>,<span class="p1">(<span class="default"><a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:81"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="gcd_1.html#E3:81"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="gcd_1.html#E2:81_1_1_3"><i><font color="Green" title="E65">A12</font></i></a>, <a class="txt" href="gcd_1.html#E3:81_1_1_3"><i><font color="Green" title="E66">A13</font></i></a>, <a class="txt" href="gcd_1.html#E8:81_1_1_3"><i><font color="Green" title="E70">A17</font></i></a>, <a class="ref" href="gcd_1.html#T42" target="_self" title="GCD_1:th.42">Th42</a></i>;<br/><a NAME="E11:81_1_1_3"/><i><font color="Green" title="E72">A19</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c6">s2</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></i>;<br/><a NAME="E12:81_1_1_3"/><i><font color="Green" title="E73">A20</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:81"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="gcd_1.html#T34" target="_self" title="GCD_1:th.34">Th34</a></i>;<br/><a NAME="E13:81_1_1_3"/><b>then </b><i><font color="Green" title="E74">A21</font></i>: 
<font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:81"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="gcd_1.html#E11:81_1_1_3"><i><font color="Green" title="E72">A19</font></i></a>, <a class="ref" href="gcd_1.html#T8" target="_self" title="GCD_1:th.8">Th8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">zz</font> =  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Amp</font> <b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">s2</font> = <font color="Maroon" title="c6">s2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Amp</font> <b>by </b><i><a class="txt" href="gcd_1.html#E1:81"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></i>;<br/><a NAME="E16:81_1_1_3"/>
<font color="Maroon" title="c8">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c7">zz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Amp</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:81"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="gcd_1.html#E11:81_1_1_3"><i><font color="Green" title="E72">A19</font></i></a>, <a class="txt" href="gcd_1.html#E12:81_1_1_3"><i><font color="Green" title="E73">A20</font></i></a>, <a class="ref" href="gcd_1.html#T27" target="_self" title="GCD_1:th.27">Th27</a></i>;<br/><b>hence </b><a NAME="E17:81_1_1_3"/>
 <a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font>, <a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Maroon" title="c2">Amp</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:81_1_1_3"><i><font color="Green" title="E66">A13</font></i></a>, <a class="txt" href="gcd_1.html#E10:81_1_1_3"><i><font color="Green" title="E71">A18</font></i></a>, <a class="txt" href="gcd_1.html#E13:81_1_1_3"><i><font color="Green" title="E74">A21</font></i></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="E64">A22</font></i>: 
( <font color="Maroon" title="c4">r2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font>, <a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Maroon" title="c2">Amp</font></span><br/><div class="add"><a NAME="E2:81_1_1_4"/><b>then </b><i><font color="Green" title="E65">A23</font></i>: 
 <a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>
 <b>by </b><i><a class="ref" href="gcd_1.html#D18" target="_self" title="GCD_1:def.18">Def18</a></i>;<br/><a NAME="E3:81_1_1_4"/><i><font color="Green" title="E66">A24</font></i>: 
 <a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:81"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="gcd_1.html#E1:81_1_1_4"><i><font color="Green" title="E64">A22</font></i></a>, <a class="ref" href="gcd_1.html#D19" target="_self" title="GCD_1:def.19">Def19</a></i>;<br/><a NAME="E4:81_1_1_4"/><i><font color="Green" title="E67">A25</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:81_1_1_4"><i><font color="Green" title="E64">A22</font></i></a>, <a class="ref" href="gcd_1.html#T33" target="_self" title="GCD_1:th.33">Th33</a></i>;<br/><a NAME="E5:81_1_1_4"/><b>then </b><i><font color="Green" title="E68">A26</font></i>: 
<font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:81_1_1_4"><i><font color="Green" title="E64">A22</font></i></a>, <a class="ref" href="gcd_1.html#T9" target="_self" title="GCD_1:th.9">Th9</a></i>;<br/><a NAME="E6:81_1_1_4"/><i><font color="Green" title="E69">A27</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">s1</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></i>;<br/><a NAME="E7:81_1_1_4"/><b>then </b>
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">r1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">s1</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T7" target="_self" title="GCD_1:th.7">Th7</a></i>;<br/><b>then </b><i><font color="Green" title="E70">A28</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> = 
<font color="Maroon" title="c3">r1</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gcd_1.html#E4:81"><i><font color="Green" title="E63">A4</font></i></a>, <a class="txt" href="gcd_1.html#E6:81_1_1_4"><i><font color="Green" title="E69">A27</font></i></a>, <a class="ref" href="gcd_1.html#T11" target="_self" title="GCD_1:th.11">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gcd_1.html#E4:81_1_1_4"><i><font color="Green" title="E67">A25</font></i></a>, <a class="ref" href="gcd_1.html#T10" target="_self" title="GCD_1:th.10">Th10</a></i>
;<br/><a NAME="E9:81_1_1_4"/>
<font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="gcd_1.html#E5:81_1_1_4"><i><font color="Green" title="E68">A26</font></i></a>, <a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>;<br/><a NAME="E10:81_1_1_4"/><b>then </b><i><font color="Green" title="E71">A29</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>,<span class="p1">(<span class="default"><a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:81"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="gcd_1.html#E3:81"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="gcd_1.html#E2:81_1_1_4"><i><font color="Green" title="E65">A23</font></i></a>, <a class="txt" href="gcd_1.html#E3:81_1_1_4"><i><font color="Green" title="E66">A24</font></i></a>, <a class="txt" href="gcd_1.html#E8:81_1_1_4"><i><font color="Green" title="E70">A28</font></i></a>, <a class="ref" href="gcd_1.html#T42" target="_self" title="GCD_1:th.42">Th42</a></i>;<br/><a NAME="E11:81_1_1_4"/><i><font color="Green" title="E72">A30</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">r2</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></i>;<br/><a NAME="E12:81_1_1_4"/><i><font color="Green" title="E73">A31</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:81"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="gcd_1.html#T34" target="_self" title="GCD_1:th.34">Th34</a></i>;<br/><a NAME="E13:81_1_1_4"/><b>then </b><i><font color="Green" title="E74">A32</font></i>: 
<font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:81"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="gcd_1.html#E11:81_1_1_4"><i><font color="Green" title="E72">A30</font></i></a>, <a class="ref" href="gcd_1.html#T8" target="_self" title="GCD_1:th.8">Th8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">zz</font> =  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Amp</font> <b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">r2</font> = <font color="Maroon" title="c4">r2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Amp</font> <b>by </b><i><a class="txt" href="gcd_1.html#E1:81"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></i>;<br/><a NAME="E16:81_1_1_4"/>
<font color="Maroon" title="c8">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c7">zz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Amp</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:81"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="gcd_1.html#E11:81_1_1_4"><i><font color="Green" title="E72">A30</font></i></a>, <a class="txt" href="gcd_1.html#E12:81_1_1_4"><i><font color="Green" title="E73">A31</font></i></a>, <a class="ref" href="gcd_1.html#T27" target="_self" title="GCD_1:th.27">Th27</a></i>;<br/><b>hence </b><a NAME="E17:81_1_1_4"/>
 <a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font>, <a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Maroon" title="c2">Amp</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:81_1_1_4"><i><font color="Green" title="E66">A24</font></i></a>, <a class="txt" href="gcd_1.html#E10:81_1_1_4"><i><font color="Green" title="E71">A29</font></i></a>, <a class="txt" href="gcd_1.html#E13:81_1_1_4"><i><font color="Green" title="E74">A32</font></i></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="E64">A33</font></i>: 
( not <font color="Maroon" title="c3">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> &amp; not <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> &amp; (  not <font color="Maroon" title="c4">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> or  not <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> ) &amp; (  not <font color="Maroon" title="c6">s2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> or  not <font color="Maroon" title="c4">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> ) &amp; (  not <font color="Maroon" title="c4">r2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> or  not <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font>, <a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Maroon" title="c2">Amp</font></span><br/><div class="add"><a NAME="E2:81_1_1_5"/><b>then </b><i><font color="Green" title="E65">A34</font></i>: 
 <a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="gcd_1.html#D18" target="_self" title="GCD_1:def.18">Def18</a></i>;<br/><a NAME="E3:81_1_1_5"/><i><font color="Green" title="E66">A35</font></i>: 
 <a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p2">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:81"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="gcd_1.html#E1:81_1_1_5"><i><font color="Green" title="E64">A33</font></i></a>, <a class="ref" href="gcd_1.html#D19" target="_self" title="GCD_1:def.19">Def19</a></i>;<br/><a NAME="E4:81_1_1_5"/><b>then </b><i><font color="Green" title="E67">A36</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>,<span class="p1">(<span class="default"><a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:81"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="gcd_1.html#E3:81"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="gcd_1.html#E2:81_1_1_5"><i><font color="Green" title="E65">A34</font></i></a>, <a class="ref" href="gcd_1.html#T42" target="_self" title="GCD_1:th.42">Th42</a></i>;<br/><a NAME="E5:81_1_1_5"/><i><font color="Green" title="E68">A37</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c6">s2</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></i>;<br/><a NAME="E6:81_1_1_5"/><i><font color="Green" title="E69">A38</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:81"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="gcd_1.html#T34" target="_self" title="GCD_1:th.34">Th34</a></i>;<br/><a NAME="E7:81_1_1_5"/><b>then </b><i><font color="Green" title="E70">A39</font></i>: 
<font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:81"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="gcd_1.html#E5:81_1_1_5"><i><font color="Green" title="E68">A37</font></i></a>, <a class="ref" href="gcd_1.html#T8" target="_self" title="GCD_1:th.8">Th8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">z1</font> =  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Amp</font> <b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">s2</font> = <font color="Maroon" title="c6">s2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Amp</font> <b>by </b><i><a class="txt" href="gcd_1.html#E1:81"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></i>;<br/><a NAME="E10:81_1_1_5"/><i><font color="Green" title="E71">A40</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">r2</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></i>;<br/><a NAME="E11:81_1_1_5"/><i><font color="Green" title="E72">A41</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:81"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="gcd_1.html#T34" target="_self" title="GCD_1:th.34">Th34</a></i>;<br/><a NAME="E12:81_1_1_5"/><b>then </b><i><font color="Green" title="E73">A42</font></i>: 
<font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:81"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="gcd_1.html#E10:81_1_1_5"><i><font color="Green" title="E71">A40</font></i></a>, <a class="ref" href="gcd_1.html#T8" target="_self" title="GCD_1:th.8">Th8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">z2</font> =  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c2">Amp</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Amp</font> <b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">r2</font> = <font color="Maroon" title="c4">r2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Amp</font> <b>by </b><i><a class="txt" href="gcd_1.html#E1:81"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">u</font> = <font color="Maroon" title="c8">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c7">z1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Amp</font> <b>by </b><i><a class="txt" href="gcd_1.html#E1:81"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="gcd_1.html#E5:81_1_1_5"><i><font color="Green" title="E68">A37</font></i></a>, <a class="txt" href="gcd_1.html#E6:81_1_1_5"><i><font color="Green" title="E69">A38</font></i></a>, <a class="ref" href="gcd_1.html#T27" target="_self" title="GCD_1:th.27">Th27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">v</font> = <font color="Maroon" title="c10">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c9">z2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Amp</font> <b>by </b><i><a class="txt" href="gcd_1.html#E1:81"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="gcd_1.html#E10:81_1_1_5"><i><font color="Green" title="E71">A40</font></i></a>, <a class="txt" href="gcd_1.html#E11:81_1_1_5"><i><font color="Green" title="E72">A41</font></i></a>, <a class="ref" href="gcd_1.html#T27" target="_self" title="GCD_1:th.27">Th27</a></i>;<br/><a NAME="E17:81_1_1_5"/><i><font color="Green" title="E74">A43</font></i>: 
<font color="Maroon" title="c12">v</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Amp</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:81"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="gcd_1.html#D10" target="_self" title="GCD_1:def.10">Def10</a></i>;<br/><a NAME="E18:81_1_1_5"/>
<font color="Maroon" title="c12">v</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c11">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E7:81_1_1_5"><i><font color="Green" title="E70">A39</font></i></a>, <a class="txt" href="gcd_1.html#E12:81_1_1_5"><i><font color="Green" title="E73">A42</font></i></a>, <a class="ref" href="vectsp_2.html#D5" title="VECTSP_2:def.5">VECTSP_2:def 5</a></i>;<br/><b>hence </b><a NAME="E19:81_1_1_5"/>
 <a href="gcd_1.html#K8" title="GCD_1:func.8">mult1</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font>, <a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Maroon" title="c2">Amp</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E3:81_1_1_5"><i><font color="Green" title="E66">A35</font></i></a>, <a class="txt" href="gcd_1.html#E4:81_1_1_5"><i><font color="Green" title="E67">A36</font></i></a>, <a class="txt" href="gcd_1.html#E17:81_1_1_5"><i><font color="Green" title="E74">A43</font></i></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="GCD_1:func.8">mult1</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font>, <a href="gcd_1.html#K9" title="GCD_1:func.9">mult2</a> <font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R8" title="GCD_1:pred.8">are_normalized_wrt</a> <font color="Maroon" title="c2">Amp</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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