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

<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>, <font color="Olive" title="b2">Amp9</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="b3">x</font>, <font color="Olive" title="b4">y</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="b3">x</font>,<font color="Olive" title="b4">y</font> <a href="gcd_1.html#R5" title="GCD_1:pred.5">are_canonical_wrt</a> <font color="Olive" title="b1">Amp</font> holds <br/><font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font> <a href="gcd_1.html#R5" title="GCD_1:pred.5">are_canonical_wrt</a> <font color="Olive" title="b2">Amp9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Amp</font>, <font color="Maroon" title="c3">Amp9</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">x</font>, <font color="Olive" title="b2">y</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">x</font>,<font color="Olive" title="b2">y</font> <a href="gcd_1.html#R5" title="GCD_1:pred.5">are_canonical_wrt</a> <font color="Maroon" title="c2">Amp</font> holds <br/><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="gcd_1.html#R5" title="GCD_1:pred.5">are_canonical_wrt</a> <font color="Maroon" title="c3">Amp9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</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="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="gcd_1.html#R5" title="GCD_1:pred.5">are_canonical_wrt</a> <font color="Maroon" title="c2">Amp</font> implies <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="gcd_1.html#R5" title="GCD_1:pred.5">are_canonical_wrt</a> <font color="Maroon" title="c3">Amp9</font> )</span><br/>





<a NAME="E1:71"/>
<span class="p1">(<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> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D19" title="VECTSP_1:def.19">VECTSP_1:def 19</a></span>;<br/>
<a NAME="E2:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A1</font></span>: 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D1" target="_self" title="GCD_1:def.1">Def1</a></span>;<br/>
<a NAME="E3:71"/>
<span class="p1">(<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> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D19" title="VECTSP_1:def.19">VECTSP_1:def 19</a></span>;<br/>
<a NAME="E4:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A2</font></span>: 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D1" target="_self" title="GCD_1:def.1">Def1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:71"/>
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="gcd_1.html#R5" title="GCD_1:pred.5">are_canonical_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"> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="gcd_1.html#R5" title="GCD_1:pred.5">are_canonical_wrt</a> <font color="Maroon" title="c3">Amp9</font></span><br/>

<a NAME="E6:71"/><span class="kw">then </span>
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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#D13" target="_self" title="GCD_1:def.13">Def13</a></span>;<br/>
<a NAME="E7:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A3</font></span>: 
for <font color="Olive" title="b1">z</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">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b1">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">y</font> holds <br/><font color="Olive" title="b1">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</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#D12" target="_self" title="GCD_1:def.12">Def12</a></span>;<br/>
<a NAME="E8:71"/>
 <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="c3">Amp9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D8" target="_self" title="GCD_1:def.8">Def8</a></span>;<br/>
<a NAME="E9:71"/><span class="kw">then </span>
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">Amp9</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="#E7:71"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="txt" href="#E4:71"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:71"/>
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="gcd_1.html#R5" title="GCD_1:pred.5">are_canonical_wrt</a> <font color="Maroon" title="c3">Amp9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D13" target="_self" title="GCD_1:def.13">Def13</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>
