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<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Ring</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">RightMod</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b2">S</font>, <font color="Olive" title="b3">T</font> being   <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="Olive" title="b1">V</font> holds   <a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="Olive" title="b3">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">S</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Olive" title="b2">S</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">RightMod</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font>, <font color="Olive" title="b2">T</font> being   <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> holds   <a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="Olive" title="b2">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b1">S</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Olive" title="b1">S</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">S</font>, <font color="Maroon" title="c4">T</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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Maroon" title="c3">S</font></span>)</span></span><br/>







<a NAME="E1:15"/>
<font color="Maroon" title="c4">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">S</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" onmouseover="rs('xboole_1/T79')" onmouseout="rh()">XBOOLE_1:79</a></span>;<br/>
<a NAME="E2:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c2">V</font>
 
;<br/>
<a NAME="E3:15"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" onmouseover="rs('xboole_1/T39')" onmouseout="rh()">XBOOLE_1:39</a></span>;<br/>
<a NAME="E4:15"/><span class="kw">then </span>
 <a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">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> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0: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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_4.html#T10" target="_self" onmouseover="rs('rmod_4/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">then </span> <a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">S</font></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"><font color="Maroon" title="c4">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">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> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="rmod_4.html#T5" target="_self" onmouseover="rs('rmod_4/T5')" onmouseout="rh()">Th5</a></span>
<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="c4">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">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> <font color="Maroon" title="c3">S</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T18" onmouseover="rs('vectsp_1/T18')" onmouseout="rh()">VECTSP_1:18</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:15"/>
 <a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">T</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Maroon" title="c3">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#T61" onmouseover="rs('rlsub_2/T61')" onmouseout="rh()">RLSUB_2:61</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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