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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="group_1.html#V7">commutative</a> non <a href="vectsp_1.html#V10">degenerated</a> <a href="vectsp_2.html#V2">domRing-like</a> <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>given </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K9" target="_self">Quot.</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:18"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:18"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K8" target="_self">QClass.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="quofield.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E4:18"/><i><font color="Green">E14</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="quofield.html#E1:18"><i><font color="Green">E12</font></i></a>, <a class="txt" href="quofield.html#E3:18"><i><font color="Green">E13</font></i></a>, <a class="ref" href="quofield.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E5:18"/><i><font color="Green">E15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="quofield.html#E1:18"><i><font color="Green">E12</font></i></a>, <a class="txt" href="quofield.html#E3:18"><i><font color="Green">E13</font></i></a>, <a class="ref" href="quofield.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E6:18"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K3" target="_self">`2</a>  <a href="gcd_1.html#R2">divides</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D1">GCD_1:def 1</a></i>;<br/>
<a NAME="E7:18"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K3" target="_self">`2</a>  <a href="gcd_1.html#R2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T7">GCD_1:7</a></i>;<br/>
<a NAME="E8:18"/>
<font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K3" target="_self">`2</a>  <a href="gcd_1.html#R2">divides</a> <font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K3" target="_self">`2</a> 
 
;<br/>
<a NAME="E9:18"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K3" target="_self">`2</a>  <a href="gcd_1.html#R2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T7">GCD_1:7</a></i>;<br/>
<a NAME="E10:18"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K3" target="_self">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="quofield.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="gcd_1.html#K1">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="quofield.html#E5:18"><i><font color="Green">E15</font></i></a>, <a class="txt" href="quofield.html#E6:18"><i><font color="Green">E16</font></i></a>, <a class="ref" href="gcd_1.html#D4">GCD_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="gcd_1.html#K1">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="quofield.html#E6:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="quofield.html#E7:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="quofield.html#E10:18"><i><font color="Green">E19</font></i></a>, <a class="ref" href="gcd_1.html#T11">GCD_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="gcd_1.html#K1">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="quofield.html#E4:18"><i><font color="Green">E14</font></i></a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="gcd_1.html#K1">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="gcd_1.html#K1">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="quofield.html#E9:18"><i><font color="Green">E18</font></i></a>, <a class="txt" href="quofield.html#E10:18"><i><font color="Green">E19</font></i></a>, <a class="ref" href="gcd_1.html#T11">GCD_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="quofield.html#E10:18"><i><font color="Green">E19</font></i></a>, <a class="ref" href="gcd_1.html#T9">GCD_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
;<br/><br/>


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