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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:82"/><span class="lab"><font color="Green" title="E60">A1</font></span>: 
<font color="Maroon" title="c2">Amp</font> is  <a href="gcd_1.html#V2" title="GCD_1:attr.2">multiplicative</a> 
 <span class="kw">and </span><br/><a NAME="E2:82"/><span class="lab"><font color="Green" title="E61">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:82"/><span class="lab"><font color="Green" title="E62">A3</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E4:82"/><span class="lab"><font color="Green" title="E63">A4</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></span>;<br/>
<a NAME="E5:82"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A5</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D14" target="_self" title="GCD_1:def.14">Def14</a></span>;<br/>
<a NAME="E6:82"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></span>;<br/>
<a NAME="E7:82"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A6</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D9" target="_self" title="GCD_1:def.9">Def9</a></span>;<br/>
<a NAME="E8:82"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></span>;<br/>
<a NAME="E9:82"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A7</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D9" target="_self" title="GCD_1:def.9">Def9</a></span>;<br/>
<a NAME="E10:82"/><span class="lab"><font color="Green" title="E67">A8</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></span>;<br/>
<a NAME="E11:82"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A9</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#T34" target="_self" title="GCD_1:th.34">Th34</a></span>;<br/>
<a NAME="E12:82"/><span class="lab"><font color="Green" title="E69">A10</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></span>;<br/>
<a NAME="E13:82"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A11</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D14" target="_self" title="GCD_1:def.14">Def14</a></span>;<br/>
<a NAME="E14:82"/><span class="lab"><font color="Green" title="E71">A12</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></span>;<br/>
<a NAME="E15:82"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A13</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#T34" target="_self" title="GCD_1:th.34">Th34</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:82_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="82_1_1_1"><span class="kw">case </span></a><a NAME="E1:82_1_1_1"/><span class="lab"><font color="Green" title="E73">A14</font></span>: 
( <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:82_1_1_1"/><span class="lab"><font color="Green" title="E74">A15</font></span>: 
(  <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> &amp;  <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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#T22" target="_self" title="GCD_1:th.22">Th22</a></span>;<br/><a NAME="E3:82_1_1_1"/><span class="lab"><font color="Green" title="E75">A16</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E13:82"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E9:82"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="txt" href="#E7:82"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="txt" href="#E1:82_1_1_1"><span class="lab"><font color="Green" title="E73">A14</font></span></a>, <a class="ref" href="gcd_1.html#D19" target="_self" title="GCD_1:def.19">Def19</a></span>;<br/><a NAME="E4:82_1_1_1"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#T33" target="_self" title="GCD_1:th.33">Th33</a></span>;<br/><span class="kw">hence </span><a NAME="E5:82_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82_1_1_1"><span class="lab"><font color="Green" title="E75">A16</font></span></a>, <a class="txt" href="#E2:82_1_1_1"><span class="lab"><font color="Green" title="E74">A15</font></span></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_2"><span class="kw">case </span></a><a NAME="E1:82_1_1_2"/><span class="lab"><font color="Green" title="E73">A17</font></span>: 
( <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:82_1_1_2"/><span class="lab"><font color="Green" title="E74">A18</font></span>: 
(  <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> &amp;  <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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#T22" target="_self" title="GCD_1:th.22">Th22</a></span>;<br/><a NAME="E3:82_1_1_2"/><span class="lab"><font color="Green" title="E75">A19</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E13:82"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E9:82"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="txt" href="#E1:82_1_1_2"><span class="lab"><font color="Green" title="E73">A17</font></span></a>, <a class="ref" href="gcd_1.html#D19" target="_self" title="GCD_1:def.19">Def19</a></span>;<br/><a NAME="E4:82_1_1_2"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#T33" target="_self" title="GCD_1:th.33">Th33</a></span>;<br/><span class="kw">hence </span><a NAME="E5:82_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82_1_1_2"><span class="lab"><font color="Green" title="E75">A19</font></span></a>, <a class="txt" href="#E2:82_1_1_2"><span class="lab"><font color="Green" title="E74">A18</font></span></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_3"><span class="kw">case </span></a><a NAME="E1:82_1_1_3"/><span class="lab"><font color="Green" title="E73">A20</font></span>: 
( <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:82_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A21</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#T33" target="_self" title="GCD_1:th.33">Th33</a></span>;<br/><a NAME="E3:82_1_1_3"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_1_1_3"><span class="lab"><font color="Green" title="E73">A20</font></span></a>, <a class="ref" href="gcd_1.html#T9" target="_self" title="GCD_1:th.9">Th9</a></span>;<br/><a NAME="E4:82_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A22</font></span>: 
<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#K8" title="GROUP_1:func.8">*</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></span>;<br/><a NAME="E5:82_1_1_3"/><span class="lab"><font color="Green" title="E76">A23</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></span>;<br/><a NAME="E6:82_1_1_3"/><span class="kw">then </span>
 <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#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#T7" target="_self" title="GCD_1:th.7">Th7</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A24</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">s1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:82"><span class="lab"><font color="Green" title="E72">A13</font></span></a>, <a class="txt" href="#E5:82_1_1_3"><span class="lab"><font color="Green" title="E76">A23</font></span></a>, <a class="ref" href="gcd_1.html#T11" target="_self" title="GCD_1:th.11">Th11</a></span>
<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#K8" title="GROUP_1:func.8">*</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1_1_3"><span class="lab"><font color="Green" title="E74">A21</font></span></a>, <a class="ref" href="gcd_1.html#T10" target="_self" title="GCD_1:th.10">Th10</a></span>
;<br/><a NAME="E8:82_1_1_3"/><span class="lab"><font color="Green" title="E78">A25</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E13:82"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E9:82"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="txt" href="#E7:82"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="txt" href="#E1:82_1_1_3"><span class="lab"><font color="Green" title="E73">A20</font></span></a>, <a class="ref" href="gcd_1.html#D19" target="_self" title="GCD_1:def.19">Def19</a></span>;<br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></span>;<br/><a NAME="E10:82_1_1_3"/><span class="lab"><font color="Green" title="E79">A26</font></span>: 
(  <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> &amp;  <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:82"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a>, <a class="ref" href="gcd_1.html#T34" target="_self" title="GCD_1:th.34">Th34</a></span>;<br/><a NAME="E11:82_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A27</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:82"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="ref" href="gcd_1.html#T8" target="_self" title="GCD_1:th.8">Th8</a></span>;<br/><a NAME="E12:82_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="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#K8" title="GROUP_1:func.8">*</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="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_1_1_3"><span class="lab"><font color="Green" title="E73">A20</font></span></a>, <a class="ref" href="gcd_1.html#D18" target="_self" title="GCD_1:def.18">Def18</a></span>;<br/><a NAME="E13:82_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A28</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E12:82"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="txt" href="#E10:82"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="txt" href="#E14:82"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="txt" href="#E8:82_1_1_3"><span class="lab"><font color="Green" title="E78">A25</font></span></a>, <a class="txt" href="#E7:82_1_1_3"><span class="lab"><font color="Green" title="E77">A24</font></span></a>, <a class="txt" href="#E4:82_1_1_3"><span class="lab"><font color="Green" title="E75">A22</font></span></a>, <a class="ref" href="gcd_1.html#T42" target="_self" title="GCD_1:th.42">Th42</a></span>;<br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></span>;<br/><a NAME="E15:82_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E10:82_1_1_3"><span class="lab"><font color="Green" title="E79">A26</font></span></a>, <a class="ref" href="gcd_1.html#T27" target="_self" title="GCD_1:th.27">Th27</a></span>;<br/><span class="kw">hence </span><a NAME="E16:82_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:82_1_1_3"><span class="lab"><font color="Green" title="E78">A25</font></span></a>, <a class="txt" href="#E13:82_1_1_3"><span class="lab"><font color="Green" title="E81">A28</font></span></a>, <a class="txt" href="#E11:82_1_1_3"><span class="lab"><font color="Green" title="E80">A27</font></span></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_4"><span class="kw">case </span></a><a NAME="E1:82_1_1_4"/><span class="lab"><font color="Green" title="E73">A29</font></span>: 
( <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:82_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A30</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#T33" target="_self" title="GCD_1:th.33">Th33</a></span>;<br/><a NAME="E3:82_1_1_4"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_1_1_4"><span class="lab"><font color="Green" title="E73">A29</font></span></a>, <a class="ref" href="gcd_1.html#T9" target="_self" title="GCD_1:th.9">Th9</a></span>;<br/><a NAME="E4:82_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A31</font></span>: 
<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#K8" title="GROUP_1:func.8">*</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></span>;<br/><a NAME="E5:82_1_1_4"/><span class="lab"><font color="Green" title="E76">A32</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></span>;<br/><a NAME="E6:82_1_1_4"/><span class="kw">then </span>
 <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#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#T7" target="_self" title="GCD_1:th.7">Th7</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A33</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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#K8" title="GROUP_1:func.8">*</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:82"><span class="lab"><font color="Green" title="E68">A9</font></span></a>, <a class="txt" href="#E5:82_1_1_4"><span class="lab"><font color="Green" title="E76">A32</font></span></a>, <a class="ref" href="gcd_1.html#T11" target="_self" title="GCD_1:th.11">Th11</a></span>
<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#K8" title="GROUP_1:func.8">*</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1_1_4"><span class="lab"><font color="Green" title="E74">A30</font></span></a>, <a class="ref" href="gcd_1.html#T10" target="_self" title="GCD_1:th.10">Th10</a></span>
;<br/><a NAME="E8:82_1_1_4"/><span class="lab"><font color="Green" title="E78">A34</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E13:82"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E9:82"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="txt" href="#E7:82"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="txt" href="#E1:82_1_1_4"><span class="lab"><font color="Green" title="E73">A29</font></span></a>, <a class="ref" href="gcd_1.html#D19" target="_self" title="GCD_1:def.19">Def19</a></span>;<br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></span>;<br/><a NAME="E10:82_1_1_4"/><span class="lab"><font color="Green" title="E79">A35</font></span>: 
(  <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> &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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:82"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a>, <a class="ref" href="gcd_1.html#T34" target="_self" title="GCD_1:th.34">Th34</a></span>;<br/><a NAME="E11:82_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A36</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:82"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="gcd_1.html#T8" target="_self" title="GCD_1:th.8">Th8</a></span>;<br/><a NAME="E12:82_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="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#K8" title="GROUP_1:func.8">*</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_1_1_4"><span class="lab"><font color="Green" title="E73">A29</font></span></a>, <a class="ref" href="gcd_1.html#D18" target="_self" title="GCD_1:def.18">Def18</a></span>;<br/><a NAME="E13:82_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A37</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E12:82"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="txt" href="#E10:82"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="txt" href="#E14:82"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="txt" href="#E8:82_1_1_4"><span class="lab"><font color="Green" title="E78">A34</font></span></a>, <a class="txt" href="#E7:82_1_1_4"><span class="lab"><font color="Green" title="E77">A33</font></span></a>, <a class="txt" href="#E4:82_1_1_4"><span class="lab"><font color="Green" title="E75">A31</font></span></a>, <a class="ref" href="gcd_1.html#T42" target="_self" title="GCD_1:th.42">Th42</a></span>;<br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></span>;<br/><a NAME="E15:82_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E10:82_1_1_4"><span class="lab"><font color="Green" title="E79">A35</font></span></a>, <a class="ref" href="gcd_1.html#T27" target="_self" title="GCD_1:th.27">Th27</a></span>;<br/><span class="kw">hence </span><a NAME="E16:82_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:82_1_1_4"><span class="lab"><font color="Green" title="E78">A34</font></span></a>, <a class="txt" href="#E13:82_1_1_4"><span class="lab"><font color="Green" title="E81">A37</font></span></a>, <a class="txt" href="#E11:82_1_1_4"><span class="lab"><font color="Green" title="E80">A36</font></span></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_5"><span class="kw">case </span></a><a NAME="E1:82_1_1_5"/><span class="lab"><font color="Green" title="E73">A38</font></span>: 
( 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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">reconsider </span><font color="Maroon" title="c7">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></span>;<br/><a NAME="E4:82_1_1_5"/><span class="lab"><font color="Green" title="E74">A39</font></span>: 
(  <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> &amp;  <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:82"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a>, <a class="ref" href="gcd_1.html#T34" target="_self" title="GCD_1:th.34">Th34</a></span>;<br/><a NAME="E5:82_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A40</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:82"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="ref" href="gcd_1.html#T8" target="_self" title="GCD_1:th.8">Th8</a></span>;<br/><a NAME="E6:82_1_1_5"/><span class="lab"><font color="Green" title="E76">A41</font></span>: 
 <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#K8" title="GROUP_1:func.8">*</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E13:82"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E9:82"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="txt" href="#E7:82"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="txt" href="#E1:82_1_1_5"><span class="lab"><font color="Green" title="E73">A38</font></span></a>, <a class="ref" href="gcd_1.html#D19" target="_self" title="GCD_1:def.19">Def19</a></span>;<br/><a NAME="E7:82_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="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#K8" title="GROUP_1:func.8">*</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_1_1_5"><span class="lab"><font color="Green" title="E73">A38</font></span></a>, <a class="ref" href="gcd_1.html#D18" target="_self" title="GCD_1:def.18">Def18</a></span>;<br/><a NAME="E8:82_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A42</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E12:82"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="txt" href="#E10:82"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="txt" href="#E14:82"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="txt" href="#E6:82_1_1_5"><span class="lab"><font color="Green" title="E76">A41</font></span></a>, <a class="ref" href="gcd_1.html#T42" target="_self" title="GCD_1:th.42">Th42</a></span>;<br/><a NAME="E9:82_1_1_5"/><span class="lab"><font color="Green" title="E78">A43</font></span>: 
(  <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> &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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:82"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a>, <a class="ref" href="gcd_1.html#T34" target="_self" title="GCD_1:th.34">Th34</a></span>;<br/><a NAME="E10:82_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A44</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:82"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="gcd_1.html#T8" target="_self" title="GCD_1:th.8">Th8</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">u</font> = <font color="Maroon" title="c9">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c8">z1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Amp</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E4:82_1_1_5"><span class="lab"><font color="Green" title="E74">A39</font></span></a>, <a class="ref" href="gcd_1.html#T27" target="_self" title="GCD_1:th.27">Th27</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">v</font> = <font color="Maroon" title="c11">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c7">z2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Amp</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E9:82_1_1_5"><span class="lab"><font color="Green" title="E78">A43</font></span></a>, <a class="ref" href="gcd_1.html#T27" target="_self" title="GCD_1:th.27">Th27</a></span>;<br/><a NAME="E15:82_1_1_5"/><span class="lab"><font color="Green" title="E80">A45</font></span>: 
<font color="Maroon" title="c12">v</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82_1_1_5"><span class="lab"><font color="Green" title="E75">A40</font></span></a>, <a class="txt" href="#E10:82_1_1_5"><span class="lab"><font color="Green" title="E79">A44</font></span></a>, <a class="ref" href="vectsp_2.html#D5" title="VECTSP_2:def.5">VECTSP_2:def 5</a></span>;<br/><a NAME="E16:82_1_1_5"/>
<font color="Maroon" title="c12">v</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Amp</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="ref" href="gcd_1.html#D10" target="_self" title="GCD_1:def.10">Def10</a></span>;<br/><span class="kw">hence </span><a NAME="E17:82_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:82_1_1_5"><span class="lab"><font color="Green" title="E76">A41</font></span></a>, <a class="txt" href="#E8:82_1_1_5"><span class="lab"><font color="Green" title="E77">A42</font></span></a>, <a class="txt" href="#E15:82_1_1_5"><span class="lab"><font color="Green" title="E80">A45</font></span></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E17:82"/>
 <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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