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

<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>;<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>;<br/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E17">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>
 ;<br/>

<b>assume </b><a NAME="E2:24"/><i><font color="Green" title="E18">A2</font></i>: 
( <font color="Maroon" title="c4">c</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">c</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c4">c</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">b</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c5">d</font> = <font color="Maroon" title="c2">a</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">c</font>;<br/>
<b>set </b><font color="Maroon" title="c6">e</font> = <font color="Maroon" title="c3">b</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">c</font>;<br/>
<a NAME="E3:24"/><i><font color="Green" title="E19">A3</font></i>: 
<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="c4">c</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E1:24"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="gcd_1.html#E2:24"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="gcd_1.html#D4" target="_self" title="GCD_1:def.4">Def4</a></i>;<br/>
<a NAME="E4:24"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</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>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E1:24"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="gcd_1.html#E2:24"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="gcd_1.html#D4" target="_self" title="GCD_1:def.4">Def4</a></i>;<br/>
<a NAME="E5:24"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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="c4">c</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E3:24"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="vectsp_1.html#D12" title="VECTSP_1:def.12">VECTSP_1:def 12</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">c</font> = 
<span class="p1">(<span class="default"><span class="p2">(<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="c4">c</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">c</font></span>)</span>
<b>by </b><i><a class="txt" href="gcd_1.html#E1:24"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="gcd_1.html#E2:24"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="gcd_1.html#T11" target="_self" title="GCD_1:th.11">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="c4">c</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</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>
<b>by </b><i><a class="txt" href="gcd_1.html#E1:24"><i><font color="Green" title="E17">A1</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"><font color="Maroon" title="c2">a</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">c</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>
;<br/>
<b>hence </b><a NAME="E7:24"/>
<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="c4">c</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">c</font></span>)</span> <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="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">c</font>
 ;<br/>


</div>
