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

<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/>

<a NAME="E1:61"/><i><font color="Green">E51</font></i>: 
 <a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">c<sub>1</sub></font> is <a href="rlvect_1.html#V4">add-associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<a NAME="E1:61_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></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">c<sub>1</sub></font></span>)</span> holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="quofield.html#E55"><i><font color="Green">Lemma47</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:61_1"/>
 <a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">c<sub>1</sub></font> is <a href="rlvect_1.html#V4">add-associative</a>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:61"/><i><font color="Green">E52</font></i>: 
 <a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">c<sub>1</sub></font> is <a href="rlvect_1.html#V5">right_zeroed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>proof </b></a><div class="add">

<a NAME="E1:61_2"/>
for <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="quofield.html#E57"><i><font color="Green">Lemma49</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:61_2"/>
 <a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">c<sub>1</sub></font> is <a href="rlvect_1.html#V5">right_zeroed</a>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:61"/>
 <a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">c<sub>1</sub></font> is <a href="rlvect_1.html#V6">right_complementable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <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">c<sub>1</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <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> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="quofield.html#K19" target="_self">quotaddinv</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> as   <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">c<sub>1</sub></font></span>)</span> ;<br/>
<b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/>

<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="quofield.html#K17" target="_self">quotadd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>

<br/>.= 
 <a href="quofield.html#K13" target="_self">q0.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="quofield.html#T30" target="_self">Th30</a></i>
<br/>.= 
 <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">c<sub>1</sub></font></span>)</span>

;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:61"/>
(  <a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">c<sub>1</sub></font> is <a href="rlvect_1.html#V4">add-associative</a> &amp;  <a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">c<sub>1</sub></font> is <a href="rlvect_1.html#V5">right_zeroed</a> &amp;  <a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">c<sub>1</sub></font> is <a href="rlvect_1.html#V6">right_complementable</a> )
 <b>by </b><i><a class="txt" href="quofield.html#E1:61"><i><font color="Green">E51</font></i></a>, <a class="txt" href="quofield.html#E2:61"><i><font color="Green">E52</font></i></a></i>;<br/>


</div>
