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<div><span class="kw">theorem </span><a NAME="T15"><span class="comment"><font color="firebrick">:: HILB10_2:15</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">X</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">A</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b1">X</font><br/>  for <font color="Olive" title="b3">R</font> being   <a href="orders_1.html#NM4" title="ORDERS_1:NM.4">Order</a> of <font color="Olive" title="b1">X</font>  st <font color="Olive" title="b3">R</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Olive" title="b2">A</font> holds <br/> for <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">k</font> &amp; <font color="Olive" title="b5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">A</font> holds <br/><span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<font color="Olive" title="b3">R</font>,<span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<font color="Olive" title="b3">R</font>,<font color="Olive" title="b2">A</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Olive" title="b5">k</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<font color="Olive" title="b3">R</font>,<font color="Olive" title="b2">A</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font></div></div>
