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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="b3">c</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="b3">c</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">a</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Olive" title="b3">c</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Olive" title="b2">b</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="c4">C</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="c4">C</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c2">A</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c4">C</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">B</font> implies <font color="Maroon" title="c2">A</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c3">B</font> )</span><br/>



<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E22">A1</font></i>: 
( <font color="Maroon" title="c4">C</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="c4">C</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c2">A</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c4">C</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">B</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="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c3">B</font></span><br/>

<a NAME="E2:31"/><b>then </b>
<font color="Maroon" title="c4">C</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c2">A</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">C</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">B</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="E3:31"/><b>then </b><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="txt" href="#E1:31"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="gcd_1.html#T15" target="_self" title="GCD_1:th.15">Th15</a></i>;<br/>
<a NAME="E4:31"/>
<font color="Maroon" title="c4">C</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="c4">C</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="#E1:31"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="gcd_1.html#D3" target="_self" title="GCD_1:def.3">Def3</a></i>;<br/>
<a NAME="E5:31"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E1:31"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="gcd_1.html#T15" target="_self" title="GCD_1:th.15">Th15</a></i>;<br/>
<b>hence </b><a NAME="E6:31"/>
<font color="Maroon" title="c2">A</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="#E3:31"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="gcd_1.html#D3" target="_self" title="GCD_1:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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