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<b>let </b><font color="Maroon" title="c1">I</font> be  non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="group_1.html#V6" title="GROUP_1:attr.6">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">u</font>, <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</font> be    <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>;<br/>



<b>consider </b><font color="Maroon" title="c5">x</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="E2:29"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="quofield.html#D5" target="_self" title="QUOFIELD:def.5">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">y</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="E4:29"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="ref" href="quofield.html#D5" target="_self" title="QUOFIELD:def.5">Def5</a></i>;<br/>
<i><font color="Green" title="E24">A3</font></i>:  <a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> = 
 <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <span class="p1">(<span class="default"><a href="quofield.html#K6" title="QUOFIELD:func.6">padd</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span>
<b>by </b><i><a class="txt" href="quofield.html#E2:29"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="quofield.html#E4:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="quofield.html#T11" target="_self" title="QUOFIELD:th.11">Th11</a></i>
<br/>.= 
 <a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">u</font>
<b>by </b><i><a class="txt" href="quofield.html#E2:29"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="quofield.html#E4:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="quofield.html#T11" target="_self" title="QUOFIELD:th.11">Th11</a></i>
;<br/>
<b>consider </b><font color="Maroon" title="c7">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="E7:29"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c7">z</font>
 <b>by </b><i><a class="ref" href="quofield.html#D5" target="_self" title="QUOFIELD:def.5">Def5</a></i>;<br/>
 <a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> <font color="Maroon" title="c2">u</font>,<span class="p1">(<span class="default"><a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span> = 
 <a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> <span class="p1">(<span class="default"><a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <span class="p2">(<span class="default"><a href="quofield.html#K6" title="QUOFIELD:func.6">padd</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="quofield.html#E2:29"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="quofield.html#E4:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="quofield.html#E7:29"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="quofield.html#T11" target="_self" title="QUOFIELD:th.11">Th11</a></i>
<br/>.= 
 <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <span class="p1">(<span class="default"><a href="quofield.html#K6" title="QUOFIELD:func.6">padd</a> <font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default"><a href="quofield.html#K6" title="QUOFIELD:func.6">padd</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#T11" target="_self" title="QUOFIELD:th.11">Th11</a></i>
<br/>.= 
 <a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <span class="p1">(<span class="default"><a href="quofield.html#K6" title="QUOFIELD:func.6">padd</a> <span class="p2">(<span class="default"><a href="quofield.html#K6" title="QUOFIELD:func.6">padd</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span>,<font color="Maroon" title="c7">z</font></span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#T4" target="_self" title="QUOFIELD:th.4">Th4</a></i>
<br/>.= 
 <a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> <span class="p1">(<span class="default"><a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <span class="p2">(<span class="default"><a href="quofield.html#K6" title="QUOFIELD:func.6">padd</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="quofield.html#K12" title="QUOFIELD:func.12">QClass.</a> <font color="Maroon" title="c7">z</font></span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#T11" target="_self" title="QUOFIELD:th.11">Th11</a></i>
<br/>.= 
 <a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> <span class="p1">(<span class="default"><a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font></span>)</span>,<font color="Maroon" title="c4">w</font>
<b>by </b><i><a class="txt" href="quofield.html#E2:29"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="quofield.html#E4:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="quofield.html#E7:29"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="quofield.html#T11" target="_self" title="QUOFIELD:th.11">Th11</a></i>
;<br/>
<b>hence </b><a NAME="E9:29"/>
(  <a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> <font color="Maroon" title="c2">u</font>,<span class="p1">(<span class="default"><a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> <span class="p1">(<span class="default"><a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font></span>)</span>,<font color="Maroon" title="c4">w</font> &amp;  <a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K10" title="QUOFIELD:func.10">qadd</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">u</font> )
 <b>by </b><i><a class="txt" href="quofield.html#E5:29"><i><font color="Green" title="E24">A3</font></i></a></i>;<br/>


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