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<b>let </b><font color="Maroon" title="c1">R</font> be  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_2.html#V1" title="VECTSP_2:attr.1">domRing-like</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</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">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="b1">a</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> &amp; <font color="Olive" title="b1">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">b</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Olive" title="b1">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b3">c</font> holds <br/><font color="Olive" title="b2">b</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Olive" title="b3">c</font></span><br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">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="c2">A</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> &amp; <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">B</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">C</font> implies <font color="Maroon" title="c3">B</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">C</font> )</span><br/>



<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E19">A1</font></i>: 
( <font color="Maroon" title="c2">A</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> &amp; <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">B</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">B</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">C</font></span><br/>

<b>then consider </b><font color="Maroon" title="c5">D</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E3:27"/><i><font color="Green" title="E20">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">C</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="E4:27"/><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">C</font>
 
<b>by </b><i><a class="txt" href="#E3:27"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>;<br/>
<a NAME="E5:27"/><i><font color="Green" title="E22">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">D</font></span>)</span></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">D</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<a NAME="E1:27_1"/>
<font color="Maroon" title="c2">A</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T6" target="_self" title="GCD_1:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E2:27_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">D</font></span>)</span></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:27"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="gcd_1.html#T11" target="_self" title="GCD_1:th.11">Th11</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="E6:27"/><i><font color="Green" title="E23">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<a NAME="E1:27_2"/>
<font color="Maroon" title="c2">A</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">C</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T6" target="_self" title="GCD_1:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E2:27_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">C</font>
 <b>by </b><i><a class="txt" href="#E1:27"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="gcd_1.html#T11" target="_self" title="GCD_1:th.11">Th11</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>
<font color="Maroon" title="c3">B</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">D</font> = 
<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="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">D</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19" title="VECTSP_1:def.19">VECTSP_1:def 19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">C</font>
<b>by </b><i><a class="txt" href="#E1:27"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="#E4:27"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="#E5:27"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="#E6:27"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="gcd_1.html#T9" target="_self" title="GCD_1:th.9">Th9</a></i>
<br/>.= 
<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="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">C</font>
<b>by </b><i><a class="txt" href="#E1:27"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="gcd_1.html#T9" target="_self" title="GCD_1:th.9">Th9</a></i>
<br/>.= 
<font color="Maroon" title="c4">C</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19" title="VECTSP_1:def.19">VECTSP_1:def 19</a></i>
;<br/>
<b>hence </b><a NAME="E8:27"/>
<font color="Maroon" title="c3">B</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">C</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D1" target="_self" title="GCD_1:def.1">Def1</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>
