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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</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">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="c1">V</font> holds   <a href="rlvect_2.html#K2" title="RLVECT_2:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">T</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</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="rlvect_2.html#K2" title="RLVECT_2:func.2">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="rlvect_2.html#K2" title="RLVECT_2:func.2">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b1">S</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font>, <font color="Maroon" title="c3">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="c1">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="rlvect_2.html#K2" title="RLVECT_2:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K2" title="RLVECT_2:func.2">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="c2">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="rlvect_2.html#K2" title="RLVECT_2:func.2">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="c2">S</font></span>)</span></span>)</span></span><br/>



<a NAME="E1:18"/>
<font color="Maroon" title="c3">T</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">S</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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="c2">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T101" onmouseover="rs('xboole_1/T101')" onmouseout="rh()">XBOOLE_1:101</a></span>;<br/>
<span class="kw">hence </span> <a href="rlvect_2.html#K2" title="RLVECT_2:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c2">S</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K2" title="RLVECT_2:func.2">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="c2">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="rlvect_2.html#K2" title="RLVECT_2:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<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="c2">S</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="c2">S</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T16" target="_self" onmouseover="rs('rlvect_2/T16')" onmouseout="rh()">Th16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K2" title="RLVECT_2:func.2">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="c2">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="rlvect_2.html#K2" title="RLVECT_2:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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="c2">S</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" onmouseover="rs('xboole_1/T16')" onmouseout="rh()">XBOOLE_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K2" title="RLVECT_2:func.2">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="c2">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="rlvect_2.html#K2" title="RLVECT_2:func.2">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="c2">S</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T21" onmouseover="rs('xboole_1/T21')" onmouseout="rh()">XBOOLE_1:21</a></span>
;<br/> <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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