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<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="gcd_1.html#NM3" title="GCD_1:NM.3">gcdDomain</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Amp</font> being   <a href="gcd_1.html#M2" title="GCD_1:mode.2">AmpleSet</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</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="b4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b1">Amp</font> &amp; <font color="Olive" title="b4">c</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> holds <br/> <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Olive" title="b4">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">b</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Olive" title="b4">c</font></span>)</span>,<font color="Olive" title="b1">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><br/><b>let </b><font color="Maroon" title="c2">Amp</font> be    <a href="gcd_1.html#M2" title="GCD_1:mode.2">AmpleSet</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</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">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Maroon" title="c2">Amp</font> &amp; <font color="Olive" title="b3">c</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> holds <br/> <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Olive" title="b3">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Olive" title="b3">c</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><br/><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c5">C</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font> &amp; <font color="Maroon" title="c5">C</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> implies  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">C</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><br/>





<b>assume </b><a NAME="E1:65"/><i><font color="Green" title="E48">A1</font></i>: 
( <font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">Amp</font> &amp; <font color="Maroon" title="c5">C</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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">C</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><br/>

<b>set </b><font color="Maroon" title="c6">A1</font> = <font color="Maroon" title="c3">A</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">C</font>;<br/>
<b>set </b><font color="Maroon" title="c7">B1</font> = <font color="Maroon" title="c4">B</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">C</font>;<br/>
<a NAME="E2:65"/>
<font color="Maroon" title="c5">C</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E1:65"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></i>;<br/>
<a NAME="E3:65"/><b>then </b><i><font color="Green" title="E49">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E1:65"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="gcd_1.html#D4" target="_self" title="GCD_1:def.4">Def4</a></i>;<br/>
<a NAME="E4:65"/>
<font color="Maroon" title="c5">C</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E1:65"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></i>;<br/>
<a NAME="E5:65"/><b>then </b><i><font color="Green" title="E50">A3</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<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">A</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E1:65"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="gcd_1.html#E3:65"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="gcd_1.html#D4" target="_self" title="GCD_1:def.4">Def4</a></i>;<br/>
<a NAME="E6:65"/>
 <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">A</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</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"><font color="Maroon" title="c3">A</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font></span>)</span>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T37" target="_self" title="GCD_1:th.37">Th37</a></i>;<br/>
<a NAME="E7:65"/><b>then </b>
<font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <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="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c5">C</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</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"><font color="Maroon" title="c3">A</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font></span>)</span>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E1:65"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="gcd_1.html#E5:65"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>;<br/>
<a NAME="E8:65"/><b>then </b><i><font color="Green" title="E51">A4</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E1:65"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="gcd_1.html#T19" target="_self" title="GCD_1:th.19">Th19</a></i>;<br/>
<a NAME="E9:65"/><i><font color="Green" title="E52">A5</font></i>: 
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Amp</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></i>;<br/>
<a NAME="E10:65"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Amp</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D8" target="_self" title="GCD_1:def.8">Def8</a></i>;<br/>
<b>hence </b><a NAME="E11:65"/>
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">Amp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E8:65"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="gcd_1.html#E9:65"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="gcd_1.html#T22" target="_self" title="GCD_1:th.22">Th22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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