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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="Red">::  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>, <font color="Olive" title="b3">b</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>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b1">Amp</font> <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> iff ( <font color="Olive" title="b2">a</font> <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> &amp; <font color="Olive" title="b3">b</font> <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><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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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>,<font color="Olive" title="b2">b</font>,<font color="Maroon" title="c2">Amp</font> <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> iff ( <font color="Olive" title="b1">a</font> <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> &amp; <font color="Olive" title="b2">b</font> <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><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</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"> (  <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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> iff ( <font color="Maroon" title="c3">A</font> <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> &amp; <font color="Maroon" title="c4">B</font> <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><br/>





<div><span class="lab"><font color="Green" title="E43">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:61_1"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
 <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="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="Red">::  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>,<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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> implies ( <font color="Maroon" title="c3">A</font> <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> &amp; <font color="Maroon" title="c4">B</font> <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><br/><a NAME="E2:61_1"/><span class="kw">then </span>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</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">B</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="E3:61_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A3</font></span>: 
 ex <font color="Olive" title="b1">E</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <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="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</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="E4:61_1"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</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="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="gcd_1.html#D12" target="_self" title="GCD_1:def.12">Def12</a></span>;<br/><a NAME="E5:61_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">D</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <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="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</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">hence </span><a NAME="E6:61_1"/>
(  <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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> implies ( <font color="Maroon" title="c3">A</font> <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> &amp; <font color="Maroon" title="c4">B</font> <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="txt" href="#E3:61_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</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><span class="kw">end;</span></div>
<a NAME="E2:61"/>
( <font color="Maroon" title="c3">A</font> <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> &amp; <font color="Maroon" title="c4">B</font> <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> implies  <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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61_2"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
<font color="Maroon" title="c3">A</font> <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">and </span><br/><a NAME="E2:61_2"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
<font color="Maroon" title="c4">B</font> <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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>,<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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span><br/>

<a NAME="E3:61_2"/>
 <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#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="#E2:61_2"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="gcd_1.html#T31" target="_self" title="GCD_1:th.31">Th31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:61_2"/>
 <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="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="txt" href="#E1:61_2"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="gcd_1.html#T25" target="_self" title="GCD_1:th.25">Th25</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:61"/>
(  <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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> iff ( <font color="Maroon" title="c3">A</font> <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> &amp; <font color="Maroon" title="c4">B</font> <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="txt" href="#E1:61"><span class="lab"><font color="Green" title="E43">A1</font></span></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>
