<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="polyform.html#NM2" title="POLYFORM:NM.2">polyhedron</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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> implies  <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> )</span><br/>

<b>assume </b><a NAME="E1:120"/><i><font color="Green" title="E100">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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></span><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="E2:120"/><i><font color="Green" title="E101">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="hidden.html#R1" title="HIDDEN:pred.1">=</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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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="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="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="E3:120"/><i><font color="Green" title="E102">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"><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="#E1:120"><i><font color="Green" title="E100">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="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/>
<b>reconsider </b><font color="Maroon" title="c5">lastterm</font> = <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"><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="c5">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="#E2:120"><i><font color="Green" title="E101">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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<b>by </b><i><a class="txt" href="#E3:120"><i><font color="Green" title="E102">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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</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> 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="E7:120"/>
 <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
