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<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">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">G</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="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="E1:8"/><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>then </b><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">a</font></span>)</span> <a href="rlvect_1.html#K2">+</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> = 
<font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">G</font>
<b>by </b><i>, </i>
;<br/>
<a NAME="E3:8"/><b>then </b>
 <a href="rlvect_1.html#K5">-</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">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</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#T19">RLVECT_1:19</a></i>;<br/>
<b>hence </b><a NAME="E4:8"/>
<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>  <a href="rlvect_1.html#K5">-</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">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T30">RLVECT_1:30</a></i>;<br/>


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