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<b>let </b><font color="Maroon" title="c1">I</font> be  <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a> non <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">degenerated</a> <a href="vectsp_2.html#V2" title="VECTSP_2:attr.2">domRing-like</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/><b>let </b><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K1" title="QUOFIELD:func.1">Q.</a> <font color="Maroon" title="c1">I</font>;<br/>



<b>given </b><font color="Maroon" title="c4">w</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K9" title="QUOFIELD:func.9">Quot.</a> <font color="Maroon" title="c1">I</font><b> such that </b><a NAME="E1:18"/><i><font color="Green" title="E12">A1</font></i>: 
( <font color="Maroon" title="c2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">w</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">z</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K1" title="QUOFIELD:func.1">Q.</a> <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E3:18"/><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K8" title="QUOFIELD:func.8">QClass.</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="ref" href="quofield.html#D5" target="_self" title="QUOFIELD:def.5">Def5</a></i>;<br/>
<a NAME="E4:18"/><i><font color="Green" title="E14">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="quofield.html#E1:18"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="quofield.html#E3:18"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="quofield.html#D4" target="_self" title="QUOFIELD:def.4">Def4</a></i>;<br/>
<a NAME="E5:18"/><i><font color="Green" title="E15">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="quofield.html#E1:18"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="quofield.html#E3:18"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="quofield.html#D4" target="_self" title="QUOFIELD:def.4">Def4</a></i>;<br/>
<a NAME="E6:18"/><b>then </b><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D1" title="GCD_1:def.1">GCD_1:def 1</a></i>;<br/>
<a NAME="E7:18"/><b>then </b><i><font color="Green" title="E17">A6</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T7" title="GCD_1:th.7">GCD_1:7</a></i>;<br/>
<a NAME="E8:18"/>
<font color="Maroon" title="c5">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> 
 
;<br/>
<a NAME="E9:18"/><b>then </b><i><font color="Green" title="E18">A7</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="gcd_1.html#T7" title="GCD_1:th.7">GCD_1:7</a></i>;<br/>
<a NAME="E10:18"/><i><font color="Green" title="E19">A8</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="quofield.html#T2" target="_self" title="QUOFIELD:th.2">Th2</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`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" title="c5">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="quofield.html#E5:18"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="quofield.html#E6:18"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="gcd_1.html#D4" title="GCD_1:def.4">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" title="c5">z</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="quofield.html#E6:18"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="quofield.html#E7:18"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="quofield.html#E10:18"><i><font color="Green" title="E19">A8</font></i></a>, <a class="ref" href="gcd_1.html#T11" title="GCD_1:th.11">GCD_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span></span>)</span></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="quofield.html#E4:18"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">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" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="quofield.html#E9:18"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="quofield.html#E10:18"><i><font color="Green" title="E19">A8</font></i></a>, <a class="ref" href="gcd_1.html#T11" title="GCD_1:th.11">GCD_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">I</font></span>)</span>
<b>by </b><i><a class="txt" href="quofield.html#E10:18"><i><font color="Green" title="E19">A8</font></i></a>, <a class="ref" href="gcd_1.html#T9" title="GCD_1:th.9">GCD_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a> </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/><br/>


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