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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="E1:118"/><i><font color="Green" title="E99">A1</font></i>: 
 <a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<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="E2:118"/><i><font color="Green" title="E100">A2</font></i>: 
 <a href="polyform.html#K29" title="POLYFORM:func.29">alternating-f-vector</a> <font color="Maroon" title="c1">p</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="polyform.html#K3" title="POLYFORM:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="polyform.html#K3" title="POLYFORM:func.3">*&gt;</a></span> <a href="polyform.html#K6" title="POLYFORM:func.6">^</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="polyform.html#K6" title="POLYFORM:func.6">^</a> <span class="p1"><a href="polyform.html#K3" title="POLYFORM:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="polyform.html#K2" title="POLYFORM:func.2">|^</a> <span class="p3">(<span class="default"><a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span><a href="polyform.html#K3" title="POLYFORM:func.3">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="polyform.html#T82" target="_self" title="POLYFORM:th.82">Th82</a></i>;<br/>
<a NAME="E3:118"/><i><font color="Green" title="E101">A3</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="polyform.html#K2" title="POLYFORM:func.2">|^</a> <span class="p1">(<span class="default"><a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="polyform.html#E1:118"><i><font color="Green" title="E99">A1</font></i></a>, <a class="ref" href="polyform.html#T8" target="_self" title="POLYFORM:th.8">Th8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">minusone</font> =  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c5">plusone</font> = 1 as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">lastterm</font> = <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="polyform.html#K2" title="POLYFORM:func.2">|^</a> <span class="p1">(<span class="default"><a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font></span>)</span> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<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#K29" title="POLYFORM:func.29">alternating-f-vector</a> <font color="Maroon" title="c1">p</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p3"><a href="polyform.html#K3" title="POLYFORM:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">minusone</font></span><a href="polyform.html#K3" title="POLYFORM:func.3">*&gt;</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p3">(<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></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p2"><a href="polyform.html#K3" title="POLYFORM:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">lastterm</font></span><a href="polyform.html#K3" title="POLYFORM:func.3">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="polyform.html#E2:118"><i><font color="Green" title="E100">A2</font></i></a>, <a class="ref" href="polyform.html#T22" target="_self" title="POLYFORM:th.22">Th22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p3"><a href="polyform.html#K3" title="POLYFORM:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">minusone</font></span><a href="polyform.html#K3" title="POLYFORM:func.3">*&gt;</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p3">(<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></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
<b>by </b><i><a class="txt" href="polyform.html#E3:118"><i><font color="Green" title="E101">A3</font></i></a>, <a class="ref" href="rvsum_1.html#T103" title="RVSUM_1:th.103">RVSUM_1:103</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p3">(<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></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="ref" href="rvsum_1.html#T103" title="RVSUM_1:th.103">RVSUM_1:103</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>

;<br/>
<b>hence </b><a NAME="E8:118"/>
 <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> <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>
 ;<br/>


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