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

<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#K4" title="SUBSET_1:func.4">\/</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"><span class="p2">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Olive" title="b3">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="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> <span class="p2">(<span class="default"><font color="Olive" title="b3">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">S</font></span>)</span></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#K4" title="SUBSET_1:func.4">\/</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"><span class="p2">(<span class="default"><a href="rmod_4.html#K1" title="RMOD_4:func.1">Sum</a> <font color="Olive" title="b2">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="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> <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="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#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> <font color="Maroon" title="c4">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="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"><font color="Maroon" title="c4">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span></span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c3">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">T</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">B</font> = <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>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">Z</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">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="c4">T</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">S</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">I</font> = <font color="Maroon" title="c4">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">S</font>;<br/>
<a NAME="E1:13"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">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="c4">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T51" onmouseover="rs('xboole_1/T51')" onmouseout="rh()">XBOOLE_1:51</a></span>;<br/>
<a NAME="E2:13"/><span class="lab"><font color="Green" title="E9">A2</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#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T51" onmouseover="rs('xboole_1/T51')" onmouseout="rh()">XBOOLE_1:51</a></span>;<br/>
<a NAME="E3:13"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">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="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> <font color="Maroon" title="c4">T</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c3">S</font>
 
;<br/>
<a NAME="E4:13"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">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="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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">S</font></span>)</span> <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#T93" onmouseover="rs('xboole_1/T93')" onmouseout="rh()">XBOOLE_1:93</a></span>;<br/>
<span class="kw">then </span><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="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="c4">T</font> <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="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="c3">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">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="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></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="c4">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">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="c4">T</font> <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="txt" href="#E3:13"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="rmod_4.html#T9" target="_self" onmouseover="rs('rmod_4/T9')" onmouseout="rh()">Th9</a>, <a class="ref" href="xboole_1.html#T103" onmouseover="rs('xboole_1/T103')" onmouseout="rh()">XBOOLE_1:103</a></span>
<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="c3">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">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="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></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="c4">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">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="c4">T</font> <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#T9" target="_self" onmouseover="rs('rmod_4/T9')" onmouseout="rh()">Th9</a>, <a class="ref" href="xboole_1.html#T82" onmouseover="rs('xboole_1/T82')" onmouseout="rh()">XBOOLE_1:82</a></span>
<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="c3">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">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="c4">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="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">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></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="c4">T</font> <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="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a></span>
<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="c3">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">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="c4">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">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="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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a></span>
<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">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="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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="rmod_4.html#T9" target="_self" onmouseover="rs('rmod_4/T9')" onmouseout="rh()">Th9</a>, <a class="ref" href="xboole_1.html#T89" onmouseover="rs('xboole_1/T89')" onmouseout="rh()">XBOOLE_1:89</a></span>
<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">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="c3">S</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="rmod_4.html#T9" target="_self" onmouseover="rs('rmod_4/T9')" onmouseout="rh()">Th9</a>, <a class="ref" href="xboole_1.html#T89" onmouseover="rs('xboole_1/T89')" onmouseout="rh()">XBOOLE_1:89</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:13"/>
 <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> <font color="Maroon" title="c4">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="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"><font color="Maroon" title="c4">T</font> <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="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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