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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="firebrick">::  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">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/><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="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
 <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>
 
<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="E2:57"/>
<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#K8" title="GROUP_1:func.8">*</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T6" title="VECTSP_1:th.6">VECTSP_1:6</a></span>;<br/>
<a NAME="E3:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A2</font></span>: 
 <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>
 
<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="E4:57"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
 <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>
 
<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="E5:57"/><span class="lab"><font color="Green" title="E43">A4</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="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"><span class="kw">proof </span></a><div class="add">

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

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:57_1"/><span class="lab"><font color="Green" title="E44">A5</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:57_1"/>
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E3:57_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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="gcd_1.html#D3" target="_self" title="GCD_1:def.3">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:57_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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="ref" href="gcd_1.html#T2" target="_self" title="GCD_1:th.2">Th2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></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="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="gcd_1.html#D3" target="_self" title="GCD_1:def.3">Def3</a></span>;<br/>
<a NAME="E7:57"/><span class="kw">then </span>
 <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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E5:57"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E40">A1</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="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>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#T29" target="_self" title="GCD_1:th.29">Th30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
