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<b>let </b><font color="Maroon">R</font> be   <a href="gcd_1.html#NM3">gcdDomain</a>;<br/><b>let </b><font color="Maroon">Amp</font> be    <a href="gcd_1.html#M2" target="_self">AmpleSet</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">B</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">C</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>





<b>assume </b><a NAME="E1:65"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">Amp</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 ;<br/>

<b>set </b><font color="Maroon">A1</font> = <font color="Maroon">A</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">B1</font> = <font color="Maroon">B</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">C</font>;<br/>
<a NAME="E2:65"/>
<font color="Maroon">C</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T7" target="_self">Th7</a>, <a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E3:65"/><b>then </b><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">C</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T7" target="_self">Th7</a>, </i>;<br/>
<a NAME="E4:65"/>
<font color="Maroon">C</font> <a href="gcd_1.html#R2" target="_self">divides</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T7" target="_self">Th7</a>, <a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E5:65"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">C</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">C</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">Amp</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T7" target="_self">Th7</a>, <a class="ref" href="gcd_1.html#T8" target="_self">Th8</a>, </i>;<br/>
<a NAME="E6:65"/>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">C</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">C</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">C</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">Amp</font></span>)</span>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E7:65"/><b>then </b>
<font color="Maroon">C</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="gcd_1.html#R4" target="_self">is_associated_to</a> <font color="Maroon">C</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">Amp</font></span>)</span>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T7" target="_self">Th7</a>, <a class="ref" href="gcd_1.html#T9" target="_self">Th9</a>, <a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>;<br/>
<a NAME="E8:65"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">Amp</font> <a href="gcd_1.html#R4" target="_self">is_associated_to</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T7" target="_self">Th7</a>, <a class="txt" href="gcd_1.html#E3:65"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E9:65"/><i><font color="Green">E53</font></i>: 
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">Amp</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E10:65"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Amp</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:65"/>
 <a href="gcd_1.html#K5" target="_self">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="gcd_1.html#K1" target="_self">/</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">Amp</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#T10" target="_self">Th10</a>, <a class="ref" href="gcd_1.html#T11" target="_self">Th11</a>, </i>;<br/>


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