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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="E1:123"/><i><font color="Green" title="E103">A1</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/>

<b>set </b><font color="Maroon" title="c2">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="E2:123"/><i><font color="Green" title="E104">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<b>by </b><i><a class="txt" href="polyform.html#E1:123"><i><font color="Green" title="E103">A1</font></i></a>, <a class="ref" href="polyform.html#D27" target="_self" title="POLYFORM:def.27">Def27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">o</font> = 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c4">tw</font> = 2 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c5">th</font> = 3 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">mo</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E7:123"/><i><font color="Green" title="E105">A3</font></i>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="polyform.html#K2" title="POLYFORM:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="polyform.html#T5" target="_self" title="POLYFORM:th.5">Th5</a>, <a class="ref" href="polyform.html#T8" target="_self" title="POLYFORM:th.8">Th8</a></i>;<br/>
<a NAME="E8:123"/><i><font color="Green" title="E106">A4</font></i>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="polyform.html#K2" title="POLYFORM:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">tw</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 
<b>by </b><i><a class="ref" href="polyform.html#T6" target="_self" title="POLYFORM:th.6">Th6</a>, <a class="ref" href="polyform.html#T9" target="_self" title="POLYFORM:th.9">Th9</a></i>;<br/>
<a NAME="E9:123"/><i><font color="Green" title="E107">A5</font></i>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="polyform.html#K2" title="POLYFORM:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">th</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="polyform.html#T7" target="_self" title="POLYFORM:th.7">Th7</a>, <a class="ref" href="polyform.html#T8" target="_self" title="POLYFORM:th.8">Th8</a></i>;<br/>
<a NAME="E10:123"/><i><font color="Green" title="E108">A6</font></i>: 
<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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o</font> <a href="polyform.html#K1" title="POLYFORM:func.1">*</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polyform.html#E1:123"><i><font color="Green" title="E103">A1</font></i></a>, <a class="txt" href="polyform.html#E7:123"><i><font color="Green" title="E105">A3</font></i></a>, <a class="ref" href="polyform.html#D27" target="_self" title="POLYFORM:def.27">Def27</a></i>;<br/>
<a NAME="E11:123"/><i><font color="Green" title="E109">A7</font></i>: 
<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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">tw</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">mo</font> <a href="polyform.html#K1" title="POLYFORM:func.1">*</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">tw</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polyform.html#E1:123"><i><font color="Green" title="E103">A1</font></i></a>, <a class="txt" href="polyform.html#E8:123"><i><font color="Green" title="E106">A4</font></i></a>, <a class="ref" href="polyform.html#D27" target="_self" title="POLYFORM:def.27">Def27</a></i>;<br/>
<a NAME="E12:123"/><i><font color="Green" title="E110">A8</font></i>: 
<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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">th</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o</font> <a href="polyform.html#K1" title="POLYFORM:func.1">*</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">th</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polyform.html#E1:123"><i><font color="Green" title="E103">A1</font></i></a>, <a class="txt" href="polyform.html#E9:123"><i><font color="Green" title="E107">A5</font></i></a>, <a class="ref" href="polyform.html#D27" target="_self" title="POLYFORM:def.27">Def27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">apcson</font> = <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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">o</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">apcstw</font> = <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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">tw</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">apcsth</font> = <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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">th</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E16:123"/><i><font color="Green" title="E111">A9</font></i>: 
 <a href="polyform.html#K30" title="POLYFORM:func.30">alternating-proper-f-vector</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polyform.html#K5" title="POLYFORM:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">apcson</font>,<font color="Maroon" title="c8">apcstw</font>,<font color="Maroon" title="c9">apcsth</font></span><a href="polyform.html#K5" title="POLYFORM:func.5">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="polyform.html#E2:123"><i><font color="Green" title="E104">A2</font></i></a>, <a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/>
 <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> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">apcson</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">apcstw</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">apcsth</font>
<b>by </b><i><a class="txt" href="polyform.html#E16:123"><i><font color="Green" title="E111">A9</font></i></a>, <a class="ref" href="rvsum_1.html#T108" title="RVSUM_1:th.108">RVSUM_1:108</a></i>
<br/>.= 
<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>
<b>by </b><i><a class="txt" href="polyform.html#E10:123"><i><font color="Green" title="E108">A6</font></i></a>, <a class="txt" href="polyform.html#E11:123"><i><font color="Green" title="E109">A7</font></i></a>, <a class="txt" href="polyform.html#E12:123"><i><font color="Green" title="E110">A8</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E18:123"/>
 <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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/>


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