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

<b>let </b><font color="Maroon">F</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/>

<b>assume </b><a NAME="E1:58"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">F</font> holds <br/> ( <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">x</font> &amp; <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</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> &amp; <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> &amp; <font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="group_1.html#K1">*</a> <font color="Olive">x</font> &amp; <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="group_1.html#K1">*</a> <font color="Olive">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; ( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> implies  ex <font color="Olive">y</font> being  <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">F</font> st <font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">F</font> ) &amp;  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">F</font> &amp; <font color="Olive">x</font> <a href="group_1.html#K1">*</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#K1">*</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#K1">*</a> <font color="Olive">z</font></span>)</span> )
 ;<br/>

<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">F</font>, <font color="Maroon">y</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">F</font>, <font color="Maroon">z</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">F</font>;<br/><b>thus </b><a NAME="E1:58_1"/>
( <font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> &amp; <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> implies  ex <font color="Olive">y</font> being  <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">F</font> st <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">F</font> ) &amp;  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">F</font> &amp; <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span> )
 <b>by </b><i><a class="ref" href="mod_2.html#T4" target="_self">Th4</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> = 
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="mod_2.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="mod_2.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="mod_2.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="mod_2.html#T4" target="_self">Th4</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E3:58"/>
<font color="Maroon">F</font> is <a href="rlvect_1.html#V6">right_complementable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a><br/>

<b>take </b>
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font>
;<br/>

<b>thus </b><a NAME="E1:58_2"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="mod_2.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:58"/>
<font color="Maroon">F</font> is   <a href="vectsp_1.html#NM2">Field</a>
 <b>by </b><i><a class="ref" href="mod_2.html#T6" target="_self">Th6</a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a>, <a class="ref" href="group_1.html#D16">GROUP_1:def 16</a>, <a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a>, <a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a>, <a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a>, <a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a>, <a class="ref" href="vectsp_1.html#D20">VECTSP_1:def 20</a>, <a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/>


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