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<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="polyform.html#NM2" title="POLYFORM:NM.2">polyhedron</a>;<br/>

<b>assume </b><a NAME="E1:128"/><i><font color="Green" title="E105">A1</font></i>: 
<font color="Maroon" title="c1">p</font> is <a href="polyform.html#V5" title="POLYFORM:attr.5">simply-connected</a>
 ;<br/>

<b>assume </b><a NAME="E2:128"/><i><font color="Green" title="E106">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> 3
 ;<br/>

<a NAME="E3:128"/><i><font color="Green" title="E107">A3</font></i>: 
<font color="Maroon" title="c1">p</font> is <a href="polyform.html#V6" title="POLYFORM:attr.6">eulerian</a>
 
<b>by </b><i><a class="txt" href="polyform.html#E1:128"><i><font color="Green" title="E105">A1</font></i></a>, <a class="ref" href="polyform.html#T89" target="_self" title="POLYFORM:th.89">Th89</a></i>;<br/>
<b>set </b><font color="Maroon" title="c2">s</font> = <span class="p1">(<span class="default"><span class="p2">(<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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="polyform.html#K11" title="POLYFORM:func.11">num-polytopes</a> <font color="Maroon" title="c1">p</font>,1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <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>,2</span>)</span>;<br/>
<a NAME="E4:128"/><i><font color="Green" title="E108">A4</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="polyform.html#K11" title="POLYFORM:func.11">num-polytopes</a> <font color="Maroon" title="c1">p</font>,1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <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>,2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K30" title="POLYFORM:func.30">alternating-proper-f-vector</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polyform.html#E2:128"><i><font color="Green" title="E106">A2</font></i></a>, <a class="ref" href="polyform.html#T87" target="_self" title="POLYFORM:th.87">Th87</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">c</font> =  <a href="polyform.html#K29" title="POLYFORM:func.29">alternating-f-vector</a> <font color="Maroon" title="c1">p</font>;<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#E3:128"><i><font color="Green" title="E107">A3</font></i></a>, <a class="ref" href="polyform.html#D31" target="_self" title="POLYFORM:def.31">Def31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="polyform.html#K11" title="POLYFORM:func.11">num-polytopes</a> <font color="Maroon" title="c1">p</font>,1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="polyform.html#K11" title="POLYFORM:func.11">num-polytopes</a> <font color="Maroon" title="c1">p</font>,2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
<b>by </b><i><a class="txt" href="polyform.html#E2:128"><i><font color="Green" title="E106">A2</font></i></a>, <a class="txt" href="polyform.html#E4:128"><i><font color="Green" title="E108">A4</font></i></a>, <a class="ref" href="polyform.html#T6" target="_self" title="POLYFORM:th.6">Th6</a>, <a class="ref" href="polyform.html#T83" target="_self" title="POLYFORM:th.83">Th83</a></i>
;<br/>
<b>hence </b><a NAME="E6:128"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyform.html#K12" title="POLYFORM:func.12">num-vertices</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="polyform.html#K13" title="POLYFORM:func.13">num-edges</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="polyform.html#K14" title="POLYFORM:func.14">num-faces</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/>


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