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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> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> holds <br/> (  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Olive" title="b2">a</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Olive" title="b1">Amp</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="Olive" title="b2">a</font>,<font color="Olive" title="b1">Amp</font> &amp;  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">Amp</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="Olive" title="b2">a</font>,<font color="Olive" title="b1">Amp</font> )</span><br/><b>let </b><font color="Maroon" title="c2">Amp</font> be    <a href="gcd_1.html#M2" title="GCD_1:mode.2">AmpleSet</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> holds <br/> (  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Olive" title="b1">a</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c2">Amp</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="Olive" title="b1">a</font>,<font color="Maroon" title="c2">Amp</font> &amp;  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c2">Amp</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="Olive" title="b1">a</font>,<font color="Maroon" title="c2">Amp</font> )</span><br/><b>let </b><font color="Maroon" title="c3">A</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"> (  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c2">Amp</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="c3">A</font>,<font color="Maroon" title="c2">Amp</font> &amp;  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">Amp</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="c3">A</font>,<font color="Maroon" title="c2">Amp</font> )</span><br/>





<a NAME="E1:57"/><i><font color="Green" title="E40">A1</font></i>: 
 <a href="gcd_1.html#K4" title="GCD_1:func.4">NF</a> <font color="Maroon" title="c3">A</font>,<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="c3">A</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D9" target="_self" title="GCD_1:def.9">Def9</a></i>;<br/>
<a NAME="E2:57"/><b>then </b><i><font color="Green" title="E41">A2</font></i>: 
 <a href="gcd_1.html#K4" title="GCD_1:func.4">NF</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D3" target="_self" title="GCD_1:def.3">Def3</a></i>;<br/>
<a NAME="E3:57"/>
<span class="p1">(<span class="default"><a href="gcd_1.html#K4" title="GCD_1:func.4">NF</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">Amp</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></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>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>;<br/>
<a NAME="E4:57"/><b>then </b><i><font color="Green" title="E42">A3</font></i>: 
 <a href="gcd_1.html#K4" title="GCD_1:func.4">NF</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">Amp</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D1" target="_self" title="GCD_1:def.1">Def1</a></i>;<br/>
<a NAME="E5:57"/><i><font color="Green" title="E43">A4</font></i>: 
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="c3">A</font> &amp; <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#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</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="gcd_1.html#K4" title="GCD_1:func.4">NF</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">Amp</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">z</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="c4">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c4">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> implies <font color="Maroon" title="c4">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a>  <a href="gcd_1.html#K4" title="GCD_1:func.4">NF</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">Amp</font> )</span><br/>

<b>assume </b><a NAME="E1:57_1"/><i><font color="Green" title="E44">A5</font></i>: 
( <font color="Maroon" title="c4">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c4">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</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"> <font color="Maroon" title="c4">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a>  <a href="gcd_1.html#K4" title="GCD_1:func.4">NF</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">Amp</font></span><br/>

<a NAME="E2:57_1"/>
<font color="Maroon" title="c3">A</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a>  <a href="gcd_1.html#K4" title="GCD_1:func.4">NF</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">Amp</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="gcd_1.html#D3" target="_self" title="GCD_1:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E3:57_1"/>
<font color="Maroon" title="c4">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a>  <a href="gcd_1.html#K4" title="GCD_1:func.4">NF</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">Amp</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:57_1"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="gcd_1.html#T2" target="_self" title="GCD_1:th.2">Th2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:57"/>
 <a href="gcd_1.html#K4" title="GCD_1:func.4">NF</a> <font color="Maroon" title="c3">A</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>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D9" target="_self" title="GCD_1:def.9">Def9</a></i>;<br/>
<a NAME="E7:57"/><b>then </b>
 <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c2">Amp</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="c3">A</font>,<font color="Maroon" title="c2">Amp</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E2:57"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="gcd_1.html#E4:57"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="gcd_1.html#E5:57"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E8:57"/>
(  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c2">Amp</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="c3">A</font>,<font color="Maroon" title="c2">Amp</font> &amp;  <a href="gcd_1.html#K5" title="GCD_1:func.5">gcd</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">Amp</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="c3">A</font>,<font color="Maroon" title="c2">Amp</font> )
 <b>by </b><i><a class="ref" href="gcd_1.html#T30" target="_self" title="GCD_1:th.30">Th30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
