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

<b>let </b><font color="Maroon">S</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">G</font> be  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="rlvect_1.html#L1">LoopStr</a> ;<br/><b>let </b><font color="Maroon">w</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">S</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>;<br/>





<b>assume </b><a NAME="E1:17"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">w</font> <a href="midsp_2.html#R2" target="_self">is_atlas_of</a> <font color="Maroon">S</font>,<font color="Maroon">G</font>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">b'</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">c'</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/>









<b>assume </b><a NAME="E2:17"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font> )
 ;<br/>

<b>thus </b><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<font color="Maroon">b'</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="midsp_2.html#K1" target="_self">Double</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b'</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="midsp_2.html#K1" target="_self">Double</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K1" target="_self">Double</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
 <a href="midsp_2.html#K1" target="_self">Double</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b'</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="midsp_2.html#K1" target="_self">Double</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">b'</font></span>)</span>
<b>by </b><i>, </i>
;<br/>


</div>
