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<div class="add">

<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/>

<a NAME="E1:72"/><i><font color="Green" title="E61">A1</font></i>: 
 <a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font> is <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>proof </b></a><div class="add">

<a NAME="E1:72_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Olive" title="b3">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="quofield.html#T47" target="_self" title="QUOFIELD:th.47">Th47</a></i>;<br/>
<b>hence </b><a NAME="E2:72_1"/>
 <a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font> is <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a>
 <b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:72"/><i><font color="Green" title="E62">A2</font></i>: 
 <a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font> is <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">Field-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><b>proof </b></a><div class="add">

<a NAME="E1:72_2"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span>  st <font color="Olive" title="b1">x</font> <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> <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span> st <font color="Olive" title="b1">x</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span>
 
<b>by </b><i><a class="ref" href="quofield.html#T48" target="_self" title="QUOFIELD:th.48">Th48</a></i>;<br/>
<b>hence </b><a NAME="E2:72_2"/>
 <a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font> is <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">Field-like</a>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D20" title="VECTSP_1:def.20">VECTSP_1:def 20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:72"/><i><font color="Green" title="E63">A3</font></i>: 
 <a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_3"><b>proof </b></a><div class="add">

<a NAME="E1:72_3"/>
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span> holds  <font color="Olive" title="b1">u</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b1">u</font>
 
<b>by </b><i><a class="txt" href="quofield.html#E56"><i><font color="Green" title="E48">Lm3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:72_3"/>
 <a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:72"/><i><font color="Green" title="E64">A4</font></i>: 
not  <a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font> is <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">degenerated</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_4"><b>proof </b></a><div class="add">

<a NAME="E1:72_4"/><i><font color="Green" title="E65">A5</font></i>: 
 <a href="quofield.html#K13" title="QUOFIELD:func.13">q0.</a> <font color="Maroon" title="c1">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="quofield.html#K14" title="QUOFIELD:func.14">q1.</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="quofield.html#T21" target="_self" title="QUOFIELD:th.21">Th21</a></i>;<br/>
<a NAME="E2:72_4"/><i><font color="Green" title="E66">A6</font></i>: 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K13" title="QUOFIELD:func.13">q0.</a> <font color="Maroon" title="c1">I</font>
 
;<br/>
<a NAME="E3:72_4"/>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K14" title="QUOFIELD:func.14">q1.</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="txt" href="quofield.html#E65"><i><font color="Green" title="E55">Lm10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:72_4"/>
not  <a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font> is <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">degenerated</a>
 <b>by </b><i><a class="txt" href="quofield.html#E1:72_4"><i><font color="Green" title="E65">A5</font></i></a>, <a class="txt" href="quofield.html#E2:72_4"><i><font color="Green" title="E66">A6</font></i></a>, <a class="ref" href="vectsp_1.html#D21" title="VECTSP_1:def.21">VECTSP_1:def 21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:72"/>
 <a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font> is <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">distributive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_5"><b>proof </b></a><div class="add">

<a NAME="E1:72_5"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Olive" title="b3">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b3">z</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">z</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Olive" title="b1">x</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="quofield.html#E76"><i><font color="Green" title="E59">Lm12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:72_5"/>
 <a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font> is <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">distributive</a>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:72"/>
 <a href="quofield.html#K21" title="QUOFIELD:func.21">the_Field_of_Quotients</a> <font color="Maroon" title="c1">I</font> is  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">distributive</a> <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">Field-like</a> non <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">degenerated</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> 
 <b>by </b><i><a class="txt" href="quofield.html#E1:72"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="quofield.html#E2:72"><i><font color="Green" title="E62">A2</font></i></a>, <a class="txt" href="quofield.html#E3:72"><i><font color="Green" title="E63">A3</font></i></a>, <a class="txt" href="quofield.html#E4:72"><i><font color="Green" title="E64">A4</font></i></a></i>;<br/>


</div>
