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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E67">Th57</font></span>: <a NAME="T59"><span class="comment"><font color="firebrick">:: POLYFORM:59</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">p</font> being   <a href="polyform.html#NM2" title="POLYFORM:NM.2">polyhedron</a><br/>  for <font color="Olive" title="b2">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/>  for <font color="Olive" title="b3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b2">k</font> <a href="polyform.html#K5" title="POLYFORM:func.5">-polytopes</a> <font color="Olive" title="b1">p</font><br/>  for <font color="Olive" title="b4">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="polyform.html#K5" title="POLYFORM:func.5">-polytopes</a> <font color="Olive" title="b1">p</font><br/>  for <font color="Olive" title="b5">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="polyform.html#K14" title="POLYFORM:func.14">-chain-space</a> <font color="Olive" title="b1">p</font></span>)</span><br/>  for <font color="Olive" title="b6">m</font>, <font color="Olive" title="b7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">n</font> <a href="polyform.html#K12" title="POLYFORM:func.12">-th-polytope</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>) &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">m</font> &amp; <font color="Olive" title="b6">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="polyform.html#K8" title="POLYFORM:func.8">num-polytopes</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>) &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b7">n</font> &amp; <font color="Olive" title="b7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="polyform.html#K8" title="POLYFORM:func.8">num-polytopes</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>) &amp; <font color="Olive" title="b6">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">n</font> holds <br/><span class="p1">(<span class="default"><a href="polyform.html#K16" title="POLYFORM:func.16">incidence-sequence</a> (<font color="Olive" title="b4">e</font>,<font color="Olive" title="b5">v</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a></div></div>
