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

<b>let </b><font color="Maroon">I</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/>

<a NAME="E1:72"/><i><font color="Green">E20</font></i>: 
 <a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font> is <a href="group_1.html#V4">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">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="group_1.html#K10">*</a> <font color="Olive">y</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="group_1.html#K10">*</a> <font color="Olive">z</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:72_1"/>
 <a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font> is <a href="group_1.html#V4">associative</a>
 <b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:72"/><i><font color="Green">E23</font></i>: 
 <a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font> is <a href="vectsp_1.html#V9">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">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font></span>)</span> holds <br/> ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font></span>)</span> st <font color="Olive">x</font> <a href="group_1.html#K10">*</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:72_2"/>
 <a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font> is <a href="vectsp_1.html#V9">Field-like</a>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D20">VECTSP_1:def 20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:72"/><i><font color="Green">E24</font></i>: 
 <a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font> is <a href="rlvect_1.html#V3">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">u</font>, <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font></span>)</span> holds  <font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">u</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:72_3"/>
 <a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font> is <a href="rlvect_1.html#V3">Abelian</a>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:72"/><i><font color="Green">E25</font></i>: 
not  <a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font> is <a href="vectsp_1.html#V10">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">E34</font></i>: 
 <a href="quofield.html#K13" target="_self">q0.</a> <font color="Maroon">I</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="quofield.html#K14" target="_self">q1.</a> <font color="Maroon">I</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:72_4"/><i><font color="Green">E39</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K13" target="_self">q0.</a> <font color="Maroon">I</font>
 
;<br/>
<a NAME="E3:72_4"/>
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K14" target="_self">q1.</a> <font color="Maroon">I</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:72_4"/>
not  <a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font> is <a href="vectsp_1.html#V10">degenerated</a>
 <b>by </b><i>, , <a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:72"/>
 <a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font> is <a href="vectsp_1.html#V7">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">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font></span>)</span> holds <br/> ( <font color="Olive">x</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="group_1.html#K10">*</a> <font color="Olive">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="group_1.html#K10">*</a> <font color="Olive">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">z</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="group_1.html#K10">*</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">z</font> <a href="group_1.html#K10">*</a> <font color="Olive">x</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:72_5"/>
 <a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font> is <a href="vectsp_1.html#V7">distributive</a>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:72"/>
 <a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font> is  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V9">Field-like</a> non <a href="vectsp_1.html#V10">degenerated</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> 
 <b>by </b><i>, <a class="ref" href="quofield.html#D6" target="_self">Def6</a>, , </i>;<br/>


</div>
