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<b>let </b><font color="Maroon" title="c1">p</font> be    <a href="polyform.html#M1" title="POLYFORM:mode.1">polyhedron</a> ;<br/>

<b>assume </b><a NAME="E1:124"/><i><font color="Green" title="E104">A1</font></i>: 
<font color="Maroon" title="c1">p</font> is <a href="polyform.html#V1" title="POLYFORM:attr.1">simply-connected</a>
 ;<br/>

<b>assume </b><a NAME="E2:124"/><i><font color="Green" title="E105">A2</font></i>: 
 <a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<b>set </b><font color="Maroon" title="c2">acs</font> =  <a href="polyform.html#K29" title="POLYFORM:func.29">alternating-f-vector</a> <font color="Maroon" title="c1">p</font>;<br/>
<b>set </b><font color="Maroon" title="c3">apcs</font> =  <a href="polyform.html#K30" title="POLYFORM:func.30">alternating-proper-f-vector</a> <font color="Maroon" title="c1">p</font>;<br/>
<a NAME="E3:124"/><i><font color="Green" title="E106">A3</font></i>: 
<font color="Maroon" title="c1">p</font> is <a href="polyform.html#V2" title="POLYFORM:attr.2">eulerian</a>
 
<b>by </b><i><a class="txt" href="polyform.html#E1:124"><i><font color="Green" title="E104">A1</font></i></a>, <a class="ref" href="polyform.html#T89" target="_self" title="POLYFORM:th.89">Th89</a></i>;<br/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
 <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p1">(<span class="default"><a href="polyform.html#K29" title="POLYFORM:func.29">alternating-f-vector</a> <font color="Maroon" title="c1">p</font></span>)</span>
<b>by </b><i><a class="txt" href="polyform.html#E3:124"><i><font color="Green" title="E106">A3</font></i></a>, <a class="ref" href="polyform.html#D29" target="_self" title="POLYFORM:def.29">Def29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p2">(<span class="default"><a href="polyform.html#K30" title="POLYFORM:func.30">alternating-proper-f-vector</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2
<b>by </b><i><a class="txt" href="polyform.html#E2:124"><i><font color="Green" title="E105">A2</font></i></a>, <a class="ref" href="polyform.html#T4" target="_self" title="POLYFORM:th.4">Th4</a>, <a class="ref" href="polyform.html#T83" target="_self" title="POLYFORM:th.83">Th83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polyform.html#K11" title="POLYFORM:func.11">num-polytopes</a> <font color="Maroon" title="c1">p</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2
<b>by </b><i><a class="txt" href="polyform.html#E2:124"><i><font color="Green" title="E105">A2</font></i></a>, <a class="ref" href="polyform.html#T85" target="_self" title="POLYFORM:th.85">Th85</a></i>
;<br/>
<b>hence </b><a NAME="E5:124"/>
 <a href="polyform.html#K12" title="POLYFORM:func.12">num-vertices</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/>


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