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<div class="add">

<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM7" title="VECTSP_2:NM.7">RightMod</a> of <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c3">T</font>, <font color="Maroon" title="c4">S</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>;<br/>







<b>set </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">T</font>;<br/>
<b>set </b><font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c3">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">S</font>;<br/>
<b>set </b><font color="Maroon" title="c7">Z</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">S</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">I</font> = <font color="Maroon" title="c3">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font>;<br/>
<a NAME="E1:13"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c4">S</font>
 
;<br/>
<a NAME="E2:13"/><b>then </b><i><font color="Green" title="E8">A1</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">S</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T93" title="XBOOLE_1:th.93">XBOOLE_1:93</a>, <a class="ref" href="xboole_1.html#T103" title="XBOOLE_1:th.103">XBOOLE_1:103</a></i>;<br/>
<a NAME="E3:13"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">T</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">S</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T82" title="XBOOLE_1:th.82">XBOOLE_1:82</a></i>;<br/>
<a NAME="E4:13"/><i><font color="Green" title="E10">A3</font></i>: 
( <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">T</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T51" title="XBOOLE_1:th.51">XBOOLE_1:51</a>, <a class="ref" href="xboole_1.html#T89" title="XBOOLE_1:th.89">XBOOLE_1:89</a></i>;<br/>
<a NAME="E5:13"/><i><font color="Green" title="E11">A4</font></i>: 
( <font color="Maroon" title="c3">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">S</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T51" title="XBOOLE_1:th.51">XBOOLE_1:51</a>, <a class="ref" href="xboole_1.html#T89" title="XBOOLE_1:th.89">XBOOLE_1:89</a></i>;<br/>
<span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E2:13"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="rmod_4.html#T9" target="_self" title="RMOD_4:th.9">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">T</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E3:13"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="rmod_4.html#T9" target="_self" title="RMOD_4:th.9">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">T</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">T</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E4:13"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="rmod_4.html#T9" target="_self" title="RMOD_4:th.9">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Maroon" title="c4">S</font></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E5:13"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="rmod_4.html#T9" target="_self" title="RMOD_4:th.9">Th9</a></i>
;<br/>
<b>hence </b><a NAME="E7:13"/>
 <a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlsub_2.html#T78" title="RLSUB_2:th.78">RLSUB_2:78</a></i>;<br/>


</div>
