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

<b>let </b><font color="Maroon" title="c1">p</font> be    <a href="polyform.html#M1" title="POLYFORM:mode.1">polyhedron</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">aspcs</font> =  <a href="polyform.html#K31" title="POLYFORM:func.31">alternating-semi-proper-f-vector</a> <font color="Maroon" title="c1">p</font>;<br/>
<b>set </b><font color="Maroon" title="c4">d</font> =  <a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font>;<br/>
<b>set </b><font color="Maroon" title="c5">r</font> = <span class="p1"><a href="polyform.html#K3" title="POLYFORM:func.3">&lt;*</a><span class="default"><span class="p2">(<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="p1">(<span class="default"><a href="polyform.html#K31" title="POLYFORM:func.31">alternating-semi-proper-f-vector</a> <font color="Maroon" title="c1">p</font></span>)</span>;<br/>
<i><font color="Green" title="E69">A1</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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#K31" title="POLYFORM:func.31">alternating-semi-proper-f-vector</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="polyform.html#K31" title="POLYFORM:func.31">alternating-semi-proper-f-vector</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="polyform.html#D26" target="_self" title="POLYFORM:def.26">Def26</a></i>
<br/>.= 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
<br/>.= 
<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="nat_1.html#K2" title="NAT_1:func.2">+</a> 2

;<br/>
<a NAME="E2:87"/><i><font color="Green" title="E70">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#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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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#K31" title="POLYFORM:func.31">alternating-semi-proper-f-vector</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polyform.html#E1:87"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="polyform.html#D24" target="_self" title="POLYFORM:def.24">Def24</a></i>;<br/>
<a NAME="E3:87"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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> holds <br/><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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</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="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#K31" title="POLYFORM:func.31">alternating-semi-proper-f-vector</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:87_2"/><i><font color="Green" title="E71">A3</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 <b>and </b><br/><a NAME="E2:87_2"/><i><font color="Green" title="E72">A4</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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>
 ;<br/>

<a NAME="E3:87_2"/><i><font color="Green" title="E73">A5</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<b>by </b><i><a class="txt" href="polyform.html#E2:87_2"><i><font color="Green" title="E72">A4</font></i></a>, <a class="ref" href="polyform.html#D24" target="_self" title="POLYFORM:def.24">Def24</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:87_2_1"/>
( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="polyform.html#E1:87_2"><i><font color="Green" title="E71">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_2_1_1"><b>suppose </b></a><a NAME="E1:87_2_1_1"/><i><font color="Green" title="E74">A6</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> = 
<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="polyform.html#K2" title="POLYFORM:func.2">|^</a> 1</span>)</span> <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">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polyform.html#E3:87_2"><i><font color="Green" title="E73">A5</font></i></a>, <a class="txt" href="polyform.html#E1:87_2_1_1"><i><font color="Green" title="E74">A6</font></i></a>, <a class="ref" href="polyform.html#D24" target="_self" title="POLYFORM:def.24">Def24</a></i>
<br/>.= 
<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#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"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>
<br/>.= 
<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#K1" title="POLYFORM:func.1">*</a> 1
<b>by </b><i><a class="ref" href="polyform.html#T31" target="_self" title="POLYFORM:th.31">Th31</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1

;<br/><b>hence </b><a NAME="E3:87_2_1_1"/>
<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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</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="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#K31" title="POLYFORM:func.31">alternating-semi-proper-f-vector</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="polyform.html#E1:87_2_1_1"><i><font color="Green" title="E74">A6</font></i></a>, <a class="ref" href="finseq_1.html#T58" title="FINSEQ_1:th.58">FINSEQ_1:58</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_2_1_2"><b>suppose </b></a><a NAME="E1:87_2_1_2"/><i><font color="Green" title="E74">A7</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:87_2_1_2"/><i><font color="Green" title="E75">A8</font></i>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="txt" href="polyform.html#E1:87_2_1_2"><i><font color="Green" title="E74">A7</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">m</font> = <font color="Maroon" title="c6">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="polyform.html#E2:87_2_1_2"><i><font color="Green" title="E75">A8</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E4:87_2_1_2"/><i><font color="Green" title="E76">A9</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="txt" href="polyform.html#E2:87_2_1_2"><i><font color="Green" title="E75">A8</font></i></a></i>;<br/><a NAME="E5:87_2_1_2"/><i><font color="Green" title="E77">A10</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="txt" href="polyform.html#E4:87_2_1_2"><i><font color="Green" title="E76">A9</font></i></a>, <a class="ref" href="nat_1.html#T19" title="NAT_1:th.19">NAT_1:19</a></i>;<br/><a NAME="E6:87_2_1_2"/><i><font color="Green" title="E78">A11</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="polyform.html#E3:87_2"><i><font color="Green" title="E73">A5</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E7:87_2_1_2"/><i><font color="Green" title="E79">A12</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="polyform.html#E6:87_2_1_2"><i><font color="Green" title="E78">A11</font></i></a></i>;<br/><a NAME="E8:87_2_1_2"/><i><font color="Green" title="E80">A13</font></i>: 
<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#K31" title="POLYFORM:func.31">alternating-semi-proper-f-vector</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polyform.html#K31" title="POLYFORM:func.31">alternating-semi-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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:87_2_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="polyform.html#K3" title="POLYFORM:func.3">&lt;*</a><span class="default"><span class="p2">(<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="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/>
<b>hence </b><a NAME="E2:87_2_1_2_1"/>
<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#K31" title="POLYFORM:func.31">alternating-semi-proper-f-vector</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polyform.html#K31" title="POLYFORM:func.31">alternating-semi-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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="polyform.html#E1:87"><i><font color="Green" title="E69">A1</font></i></a>, <a class="txt" href="polyform.html#E3:87_2"><i><font color="Green" title="E73">A5</font></i></a>, <a class="txt" href="polyform.html#E1:87_2_1_2"><i><font color="Green" title="E74">A7</font></i></a>, <a class="ref" href="finseq_1.html#T37" title="FINSEQ_1:th.37">FINSEQ_1:37</a></i>;<br/>


</div><b>end;</b></div><span class="p1">(<span class="default"><a href="polyform.html#K31" title="POLYFORM:func.31">alternating-semi-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="c7">m</font> = 
<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="polyform.html#K2" title="POLYFORM:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <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="c7">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polyform.html#E5:87_2_1_2"><i><font color="Green" title="E77">A10</font></i></a>, <a class="txt" href="polyform.html#E7:87_2_1_2"><i><font color="Green" title="E79">A12</font></i></a>, <a class="ref" href="polyform.html#D26" target="_self" title="POLYFORM:def.26">Def26</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="polyform.html#K2" title="POLYFORM:func.2">|^</a> <font color="Maroon" title="c6">n</font></span>)</span> <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="c6">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span>

;<br/><b>hence </b><a NAME="E10:87_2_1_2"/>
<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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</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="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#K31" title="POLYFORM:func.31">alternating-semi-proper-f-vector</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="polyform.html#E1:87_2"><i><font color="Green" title="E71">A3</font></i></a>, <a class="txt" href="polyform.html#E3:87_2"><i><font color="Green" title="E73">A5</font></i></a>, <a class="txt" href="polyform.html#E8:87_2_1_2"><i><font color="Green" title="E80">A13</font></i></a>, <a class="ref" href="polyform.html#D24" target="_self" title="POLYFORM:def.24">Def24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:87"/>
 <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"><a href="polyform.html#K3" title="POLYFORM:func.3">&lt;*</a><span class="default"><span class="p2">(<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="p1">(<span class="default"><a href="polyform.html#K31" title="POLYFORM:func.31">alternating-semi-proper-f-vector</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="polyform.html#E2:87"><i><font color="Green" title="E70">A2</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div>
