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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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> is  <a href="polyform.html#V5" title="POLYFORM:attr.5">simply-connected</a>  &amp;  <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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyform.html#K12" title="POLYFORM:func.12">num-vertices</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="polyform.html#K13" title="POLYFORM:func.13">num-edges</a> <font color="Maroon" title="c1">p</font></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#K14" title="POLYFORM:func.14">num-faces</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">s</font> = <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/>
<span class="kw">set </span><font color="Maroon" title="c3">c</font> =  <a href="polyform.html#K28" title="POLYFORM:func.28">alternating-f-vector</a> <font color="Maroon" title="c1">p</font>;<br/>
<span class="kw">assume </span><a NAME="E1:127"/>
<font color="Maroon" title="c1">p</font> is  <a href="polyform.html#V5" title="POLYFORM:attr.5">simply-connected</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <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 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyform.html#K12" title="POLYFORM:func.12">num-vertices</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="polyform.html#K13" title="POLYFORM:func.13">num-edges</a> <font color="Maroon" title="c1">p</font></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#K14" title="POLYFORM:func.14">num-faces</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )</span><br/>

<a NAME="E2:127"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A1</font></span>: 
<font color="Maroon" title="c1">p</font> is  <a href="polyform.html#V6" title="POLYFORM:attr.6">eulerian</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyform.html#T88" target="_self" title="POLYFORM:th.88">Th89</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:127"/><span class="lab"><font color="Green" title="E104">A2</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="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyform.html#K12" title="POLYFORM:func.12">num-vertices</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="polyform.html#K13" title="POLYFORM:func.13">num-edges</a> <font color="Maroon" title="c1">p</font></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#K14" title="POLYFORM:func.14">num-faces</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span><br/>

<a NAME="E4:127"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A3</font></span>: 
<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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="polyform.html#K29" title="POLYFORM:func.29">alternating-proper-f-vector</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyform.html#T86" target="_self" title="POLYFORM:th.86">Th87</a></span>;<br/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="polyform.html#K28" title="POLYFORM:func.28">alternating-f-vector</a> <font color="Maroon" title="c1">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:127"><span class="lab"><font color="Green" title="E103">A1</font></span></a>, <a class="ref" href="polyform.html#D31" target="_self" title="POLYFORM:def.31">Def31</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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="p2">(<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>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:127"><span class="lab"><font color="Green" title="E104">A2</font></span></a>, <a class="txt" href="#E4:127"><span class="lab"><font color="Green" title="E105">A3</font></span></a>, <a class="ref" href="polyform.html#T6" target="_self" title="POLYFORM:th.6">Th6</a>, <a class="ref" href="polyform.html#T82" target="_self" title="POLYFORM:th.82">Th83</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:127"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyform.html#K12" title="POLYFORM:func.12">num-vertices</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="polyform.html#K13" title="POLYFORM:func.13">num-edges</a> <font color="Maroon" title="c1">p</font></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#K14" title="POLYFORM:func.14">num-faces</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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