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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#K6" title="GCD_1:func.6">add1</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#K7" title="GCD_1:func.7">add2</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#K6" title="GCD_1:func.6">add1</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#K7" title="GCD_1:func.7">add2</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#K6" title="GCD_1:func.6">add1</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#K7" title="GCD_1:func.7">add2</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="E1:78"/><span class="lab"><font color="Green" title="E58">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="E2:78"/><span class="lab"><font color="Green" title="E59">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="E3:78"/><span class="lab"><font color="Green" title="E60">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#K6" title="GCD_1:func.6">add1</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#K7" title="GCD_1:func.7">add2</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="E4:78"/><span class="lab"><font color="Green" title="E61">A4</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="#E2:78"><span class="lab"><font color="Green" title="E59">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="E5:78"/>
<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="#E2:78"><span class="lab"><font color="Green" title="E59">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="E6:78"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A5</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="E7:78"/>
<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="#E3:78"><span class="lab"><font color="Green" title="E60">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="E8:78"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">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="E9:78"/><span class="lab"><font color="Green" title="E64">A7</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="#E2:78"><span class="lab"><font color="Green" title="E59">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="E10:78"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A8</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="E11:78"/><span class="lab"><font color="Green" title="E66">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="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="#E3:78"><span class="lab"><font color="Green" title="E60">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="E12:78"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A10</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="E13:78"/><span class="lab"><font color="Green" title="E68">A11</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="#E3:78"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="ref" href="gcd_1.html#D15" target="_self" title="GCD_1:def.15">Def15</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_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="E1:78_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  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</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> or <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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span> <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="c3">r1</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="c5">s1</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="c4">r2</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#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> &amp; <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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1_1"><span class="kw">case </span></a><a NAME="E1:78_1_1_1"/><span class="lab"><font color="Green" title="E69">A12</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>
 ; <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#K6" title="GCD_1:func.6">add1</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#K7" title="GCD_1:func.7">add2</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="E2:78_1_1_1"/><span class="kw">then </span>
 <a href="gcd_1.html#K7" title="GCD_1:func.7">add2</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:78"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E10:78"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="txt" href="#E12:78"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="ref" href="gcd_1.html#D17" target="_self" title="GCD_1:def.17">Def17</a></span>;<br/><span class="kw">hence </span><a NAME="E3:78_1_1_1"/>
 <a href="gcd_1.html#K6" title="GCD_1:func.6">add1</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#K7" title="GCD_1:func.7">add2</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="#E3:78"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="txt" href="#E6:78"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E10:78"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="txt" href="#E12:78"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="txt" href="#E1:78_1_1_1"><span class="lab"><font color="Green" title="E69">A12</font></span></a>, <a class="ref" href="gcd_1.html#D16" target="_self" title="GCD_1:def.16">Def16</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="78_1_1_2"><span class="kw">case </span></a><a NAME="E1:78_1_1_2"/><span class="lab"><font color="Green" title="E69">A13</font></span>: 
<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#K6" title="GCD_1:func.6">add1</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#K7" title="GCD_1:func.7">add2</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="E2:78_1_1_2"/><span class="kw">then </span>
 <a href="gcd_1.html#K7" title="GCD_1:func.7">add2</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:78"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E10:78"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="txt" href="#E12:78"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="ref" href="gcd_1.html#D17" target="_self" title="GCD_1:def.17">Def17</a></span>;<br/><span class="kw">hence </span><a NAME="E3:78_1_1_2"/>
 <a href="gcd_1.html#K6" title="GCD_1:func.6">add1</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#K7" title="GCD_1:func.7">add2</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="#E2:78"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E6:78"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E10:78"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="txt" href="#E12:78"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="txt" href="#E1:78_1_1_2"><span class="lab"><font color="Green" title="E69">A13</font></span></a>, <a class="ref" href="gcd_1.html#D16" target="_self" title="GCD_1:def.16">Def16</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="78_1_1_3"><span class="kw">case </span></a><a NAME="E1:78_1_1_3"/><span class="lab"><font color="Green" title="E69">A14</font></span>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</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>
 ; <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#K6" title="GCD_1:func.6">add1</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#K7" title="GCD_1:func.7">add2</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="E2:78_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A15</font></span>: 
 <a href="gcd_1.html#K7" title="GCD_1:func.7">add2</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="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:78"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E10:78"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="txt" href="#E12:78"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="ref" href="gcd_1.html#D17" target="_self" title="GCD_1:def.17">Def17</a></span>;<br/><a NAME="E3:78_1_1_3"/>
 <a href="gcd_1.html#K6" title="GCD_1:func.6">add1</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="c6">s2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">s1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:78"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E10:78"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="txt" href="#E12:78"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="txt" href="#E1:78_1_1_3"><span class="lab"><font color="Green" title="E69">A14</font></span></a>, <a class="ref" href="gcd_1.html#D16" target="_self" title="GCD_1:def.16">Def16</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A16</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#K6" title="GCD_1:func.6">add1</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#K7" title="GCD_1:func.7">add2</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="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">r2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">s2</font></span>)</span>,<font color="Maroon" title="c2">Amp</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78_1_1_3"><span class="lab"><font color="Green" title="E70">A15</font></span></a>, <a class="ref" href="gcd_1.html#T10" target="_self" title="GCD_1:th.10">Th10</a></span>
<br/>.= 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">s2</font></span>)</span>,<font color="Maroon" title="c2">Amp</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#T10" target="_self" title="GCD_1:th.10">Th10</a></span>
<br/>.= 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">Amp</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1_1_3"><span class="lab"><font color="Green" title="E69">A14</font></span></a>, <a class="ref" href="gcd_1.html#T10" target="_self" title="GCD_1:th.10">Th10</a></span>
<br/>.= 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>,<font color="Maroon" title="c2">Amp</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E9:78"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E11:78"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="ref" href="gcd_1.html#T41" target="_self" title="GCD_1:th.41">Th41</a></span>
<br/>.= 
 <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="#E1:78_1_1_3"><span class="lab"><font color="Green" title="E69">A14</font></span></a>, <a class="ref" href="gcd_1.html#T33" target="_self" title="GCD_1:th.33">Th33</a></span>
;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">r2</font> = <font color="Maroon" title="c4">r2</font>, <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="#E2:78"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E3:78"><span class="lab"><font color="Green" title="E60">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="E6:78_1_1_3"/>
( <font color="Maroon" title="c7">r2</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">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="c7">r2</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">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="#E1:78"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E4:78"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E13:78"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="ref" href="gcd_1.html#D10" target="_self" title="GCD_1:def.10">Def10</a>, <a class="ref" href="vectsp_2.html#D5" title="VECTSP_2:def.5">VECTSP_2:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E7:78_1_1_3"/>
 <a href="gcd_1.html#K6" title="GCD_1:func.6">add1</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#K7" title="GCD_1:func.7">add2</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="#E2:78_1_1_3"><span class="lab"><font color="Green" title="E70">A15</font></span></a>, <a class="txt" href="#E4:78_1_1_3"><span class="lab"><font color="Green" title="E71">A16</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="78_1_1_4"><span class="kw">case </span></a><a NAME="E1:78_1_1_4"/><span class="lab"><font color="Green" title="E69">A17</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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span> <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#K6" title="GCD_1:func.6">add1</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#K7" title="GCD_1:func.7">add2</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="E2:78_1_1_4"/><span class="lab"><font color="Green" title="E70">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="E3:78_1_1_4"/><span class="lab"><font color="Green" title="E71">A19</font></span>: 
 <a href="gcd_1.html#K7" title="GCD_1:func.7">add2</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="#E6:78"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E10:78"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="txt" href="#E12:78"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="txt" href="#E1:78_1_1_4"><span class="lab"><font color="Green" title="E69">A17</font></span></a>, <a class="ref" href="gcd_1.html#D17" target="_self" title="GCD_1:def.17">Def17</a></span>;<br/><a NAME="E4:78_1_1_4"/><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#K6" title="GCD_1:func.6">add1</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#K7" title="GCD_1:func.7">add2</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="E5:78_1_1_4"/>
 <a href="gcd_1.html#K6" title="GCD_1:func.6">add1</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#K7" title="GCD_1:func.7">add2</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="#E3:78_1_1_4"><span class="lab"><font color="Green" title="E71">A19</font></span></a>, <a class="txt" href="#E2:78_1_1_4"><span class="lab"><font color="Green" title="E70">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="78_1_1_5"><span class="kw">case </span></a><a NAME="E1:78_1_1_5"/>
( <font color="Maroon" title="c3">r1</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="c5">s1</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="c4">r2</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#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> &amp; <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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></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> )
 ; <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#K6" title="GCD_1:func.6">add1</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#K7" title="GCD_1:func.7">add2</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="E2:78_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A20</font></span>: 
(  <a href="gcd_1.html#K6" title="GCD_1:func.6">add1</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"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p5">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p5">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>,<font color="Maroon" title="c2">Amp</font></span>)</span> &amp;  <a href="gcd_1.html#K7" title="GCD_1:func.7">add2</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="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p5">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p5">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>,<font color="Maroon" title="c2">Amp</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:78"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E10:78"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="txt" href="#E12:78"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="ref" href="gcd_1.html#D16" target="_self" title="GCD_1:def.16">Def16</a>, <a class="ref" href="gcd_1.html#D17" target="_self" title="GCD_1:def.17">Def17</a></span>;<br/><a NAME="E3:78_1_1_5"/>
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</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="#E4:78"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="ref" href="gcd_1.html#T34" target="_self" title="GCD_1:th.34">Th34</a></span>;<br/><a NAME="E4:78_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A21</font></span>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</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#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="E5:78_1_1_5"/>
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p3">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>,<font color="Maroon" title="c2">Amp</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E9:78"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E11:78"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="ref" href="gcd_1.html#T41" target="_self" title="GCD_1:th.41">Th41</a></span>;<br/><a NAME="E6:78_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A22</font></span>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p5">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p5">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span></span>)</span> <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p0">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p0">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span> <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p0">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p0">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">r2</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>,<font color="Maroon" title="c2">Amp</font></span>)</span></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="#E4:78_1_1_5"><span class="lab"><font color="Green" title="E70">A21</font></span></a>, <a class="ref" href="gcd_1.html#T39" target="_self" title="GCD_1:th.39">Th39</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">r2</font> = <font color="Maroon" title="c4">r2</font>, <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="#E2:78"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E3:78"><span class="lab"><font color="Green" title="E60">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="E8:78_1_1_5"/><span class="lab"><font color="Green" title="E72">A23</font></span>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c7">r2</font>,<font color="Maroon" title="c8">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="c8">s2</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="c9">z2</font> =  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c7">r2</font>,<font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">r2</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p4">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c7">r2</font>,<font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c7">r2</font>,<font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>,<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="E10:78_1_1_5"/><span class="lab"><font color="Green" title="E73">A24</font></span>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c7">r2</font>,<font color="Maroon" title="c8">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="#E4:78"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="ref" href="gcd_1.html#T34" target="_self" title="GCD_1:th.34">Th34</a></span>;<br/><a NAME="E11:78_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A25</font></span>: 
<font color="Maroon" title="c9">z2</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="E12:78_1_1_5"/>
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c7">r2</font>,<font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c2">Amp</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="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">z3</font> = <font color="Maroon" title="c8">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="c7">r2</font>,<font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> 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="#E1:78"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E8:78_1_1_5"><span class="lab"><font color="Green" title="E72">A23</font></span></a>, <a class="txt" href="#E10:78_1_1_5"><span class="lab"><font color="Green" title="E73">A24</font></span></a>, <a class="ref" href="gcd_1.html#T27" target="_self" title="GCD_1:th.27">Th27</a></span>;<br/><a NAME="E14:78_1_1_5"/>
<font color="Maroon" title="c7">r2</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">z3</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="#E1:78"><span class="lab"><font color="Green" title="E58">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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">z1</font> = <font color="Maroon" title="c7">r2</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">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="c7">r2</font>,<font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Amp</font> ;<br/><a NAME="E16:78_1_1_5"/><span class="lab"><font color="Green" title="E75">A26</font></span>: 
<font color="Maroon" title="c7">r2</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">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="#E4:78"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E13:78"><span class="lab"><font color="Green" title="E68">A11</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="E17:78_1_1_5"/><span class="lab"><font color="Green" title="E76">A27</font></span>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c7">r2</font>,<font color="Maroon" title="c8">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="c7">r2</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:78_1_1_5"><span class="lab"><font color="Green" title="E72">A23</font></span></a>, <a class="ref" href="gcd_1.html#T7" target="_self" title="GCD_1:th.7">Th7</a></span>;<br/><a NAME="E18:78_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c11">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r2</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">s2</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="c7">r2</font>,<font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c2">Amp</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:78_1_1_5"><span class="lab"><font color="Green" title="E72">A23</font></span></a>, <a class="txt" href="#E10:78_1_1_5"><span class="lab"><font color="Green" title="E73">A24</font></span></a>, <a class="ref" href="gcd_1.html#T11" target="_self" title="GCD_1:th.11">Th11</a></span>;<br/><a NAME="E19:78_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A28</font></span>: 
<font color="Maroon" title="c11">z1</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="#E10:78_1_1_5"><span class="lab"><font color="Green" title="E73">A24</font></span></a>, <a class="txt" href="#E16:78_1_1_5"><span class="lab"><font color="Green" title="E75">A26</font></span></a>, <a class="txt" href="#E17:78_1_1_5"><span class="lab"><font color="Green" title="E76">A27</font></span></a>, <a class="ref" href="gcd_1.html#T8" target="_self" title="GCD_1:th.8">Th8</a></span>;<br/><a NAME="E20:78_1_1_5"/>
( <font color="Maroon" title="c9">z2</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a>  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c7">r2</font>,<font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c2">Amp</font> &amp;  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c7">r2</font>,<font color="Maroon" title="c8">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="c7">r2</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="E21:78_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A29</font></span>: 
<font color="Maroon" title="c9">z2</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c7">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#T2" target="_self" title="GCD_1:th.2">Th2</a></span>;<br/><a NAME="E22:78_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c9">z2</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c11">z1</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/><a NAME="E23:78_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A30</font></span>: 
<font color="Maroon" title="c11">z1</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c9">z2</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="#E11:78_1_1_5"><span class="lab"><font color="Green" title="E74">A25</font></span></a>, <a class="txt" href="#E19:78_1_1_5"><span class="lab"><font color="Green" title="E77">A28</font></span></a>, <a class="ref" href="gcd_1.html#T8" target="_self" title="GCD_1:th.8">Th8</a></span>;<br/><a NAME="E24:78_1_1_5"/>
<font color="Maroon" title="c11">z1</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c9">z2</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="#E1:78"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E21:78_1_1_5"><span class="lab"><font color="Green" title="E78">A29</font></span></a>, <a class="txt" href="#E11:78_1_1_5"><span class="lab"><font color="Green" title="E74">A25</font></span></a>, <a class="ref" href="gcd_1.html#T7" target="_self" title="GCD_1:th.7">Th7</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="E25:78_1_1_5"/>
 <a href="gcd_1.html#K6" title="GCD_1:func.6">add1</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#K7" title="GCD_1:func.7">add2</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="#E2:78_1_1_5"><span class="lab"><font color="Green" title="E69">A20</font></span></a>, <a class="txt" href="#E6:78_1_1_5"><span class="lab"><font color="Green" title="E71">A22</font></span></a>, <a class="txt" href="#E23:78_1_1_5"><span class="lab"><font color="Green" title="E79">A30</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="E15:78"/>
 <a href="gcd_1.html#K6" title="GCD_1:func.6">add1</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#K7" title="GCD_1:func.7">add2</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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