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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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 implies  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">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> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c2">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/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">th</font> = 3 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">tw</font> = 2 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">o</font> = 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<span class="kw">assume </span><a NAME="E5:123"/><span class="lab"><font color="Green" title="E101">A1</font></span>: 
 <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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">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></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">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="E6:123"/>
<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
 
<span class="kw">by</span> <span class="lab"><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></span>;<br/>
<a NAME="E7:123"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A2</font></span>: 
<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="c2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:123"><span class="lab"><font color="Green" title="E101">A1</font></span></a>, <a class="ref" href="polyform.html#D27" target="_self" title="POLYFORM:def.27">Def27</a></span>;<br/>
<a NAME="E8:123"/>
<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">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
 
<span class="kw">by</span> <span class="lab"><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></span>;<br/>
<a NAME="E9:123"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A3</font></span>: 
<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">th</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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">th</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:123"><span class="lab"><font color="Green" title="E101">A1</font></span></a>, <a class="ref" href="polyform.html#D27" target="_self" title="POLYFORM:def.27">Def27</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">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="c3">th</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<span class="kw">reconsider </span><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/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">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="c5">o</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E13:123"/>
<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">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
 
<span class="kw">by</span> <span class="lab"><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></span>;<br/>
<a NAME="E14:123"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A4</font></span>: 
<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">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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">o</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:123"><span class="lab"><font color="Green" title="E101">A1</font></span></a>, <a class="ref" href="polyform.html#D27" target="_self" title="POLYFORM:def.27">Def27</a></span>;<br/>
<a NAME="E15:123"/>
 <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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:123"><span class="lab"><font color="Green" title="E101">A1</font></span></a>, <a class="ref" href="polyform.html#D27" target="_self" title="POLYFORM:def.27">Def27</a></span>;<br/>
<a NAME="E16:123"/><span class="kw">then </span>
 <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="c9">apcson</font>,<font color="Maroon" title="c8">apcstw</font>,<font color="Maroon" title="c7">apcsth</font></span><a href="polyform.html#K5" title="POLYFORM:func.5">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<span class="kw">then </span> <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">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="c9">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="c7">apcsth</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T108" title="RVSUM_1:th.108">RVSUM_1:108</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:123"><span class="lab"><font color="Green" title="E104">A4</font></span></a>, <a class="txt" href="#E7:123"><span class="lab"><font color="Green" title="E102">A2</font></span></a>, <a class="txt" href="#E9:123"><span class="lab"><font color="Green" title="E103">A3</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E18:123"/>
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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