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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</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#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a> <a href="vectsp_1.html#V9">Field-like</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/>

<b>thus </b><a NAME="E1:75_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="vectsp_2.html#V2">domRing-like</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:75_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:75_1_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:75_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="quofield.html#E1:75_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:75_1_1_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="quofield.html#E2:75_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="vectsp_1.html#T33">VECTSP_1:33</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a><br/>

<b>thus </b><a NAME="E2:75_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>;<br/>


</div>
