<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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:()"><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/> ( <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> iff  ex <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 <br/>( <font color="Olive" title="b3">c</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>  &amp; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) )</span><br/>

<a NAME="E1:30"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
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>  st <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> holds <br/> ex <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 <br/>( <font color="Olive" title="b3">c</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>  &amp; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">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"> ( <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> implies  ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">c</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_1"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">c</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> )</span><br/>

<a NAME="E2:30_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D3" target="_self" title="GCD_1:def.3">Def3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">C</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E4:30_1"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
<font color="Maroon" title="c3">B</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>
 <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="E5:30_1"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E22">A2</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">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E7:30_1"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_1_1_1"/>
( <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> or <font color="Maroon" title="c2">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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:30_1_1_1_1"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">c</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> )</span><br/><div class="add"><a NAME="E2:30_1_1_1_1"/>
<font color="Maroon" title="c2">A</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">D</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E7:30_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>;<br/><a NAME="E3:30_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">C</font> <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>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1_1_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="gcd_1.html#T17" target="_self" title="GCD_1:th.17">Th17</a></span>;<br/><a NAME="E4:30_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">C</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</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="E5:30_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">C</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D2" target="_self" title="GCD_1:def.2">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:30_1_1_1_1"/>
 ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">c</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1"><span class="lab"><font color="Green" title="E23">A3</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:30_1_1_1_2"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<font color="Maroon" title="c2">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">c</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> )</span><br/><div class="add"><a NAME="E2:30_1_1_1_2"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font>
 ;<br/><a NAME="E3:30_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A7</font></span>: 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D2" target="_self" title="GCD_1:def.2">Def2</a></span>;<br/><a NAME="E4:30_1_1_1_2"/>
<font color="Maroon" title="c3">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="#E4:30_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E1:30_1_1_1_2"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></span>;<br/><a NAME="E5:30_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">B</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> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1_1_2"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E6:30_1_1_1_2"/>
 ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">c</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1_1_2"><span class="lab"><font color="Green" title="E26">A7</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:30_1"/>
 ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">c</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> )
 ; <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="E2:30"/>
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>  st  ex <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 <br/>( <font color="Olive" title="b3">c</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>  &amp; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">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"> (  ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">c</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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/>

<a NAME="E1:30_2"/>
(  ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">c</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:30_2_1_1"/>
 ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">c</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">c</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">C</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E3:30_2_1_1"/><span class="lab"><font color="Green" title="E22">A8</font></span>: 
<font color="Maroon" title="c4">C</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a> 
 <span class="kw">and </span><br/><a NAME="E4:30_2_1_1"/><span class="lab"><font color="Green" title="E23">A9</font></span>: 
<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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 ;<br/><a NAME="E5:30_2_1_1"/>
<font color="Maroon" title="c4">C</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_2_1_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="ref" href="gcd_1.html#D2" target="_self" title="GCD_1:def.2">Def2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E7:30_2_1_1"/><span class="lab"><font color="Green" title="E24">A10</font></span>: 
<font color="Maroon" title="c4">C</font> <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>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</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/><font color="Maroon" title="c2">A</font> = 
<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="c4">C</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">D</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_2_1_1"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></span>
<br/>.= 
<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="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_2_1_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
;<br/><a NAME="E9:30_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A11</font></span>: 
<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>
 <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="E10:30_2_1_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="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_2_1_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <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="E11:30_2_1_1"/>
(  ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">c</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_2_1_1"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="ref" href="gcd_1.html#D3" target="_self" title="GCD_1:def.3">Def3</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="E2:30_2_1"/>
(  ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">c</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> )
 ; <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="E2:30_2"/>
(  ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">c</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> )
 ; <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="E3:30"/>
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/> ( <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> iff  ex <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 <br/>( <font color="Olive" title="b3">c</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>  &amp; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E21">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>
