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<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="polyform.html#NM2" title="POLYFORM:NM.2">polyhedron</a>;<br/><b>let </b><font color="Maroon" title="c2">k</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><b>let </b><font color="Maroon" title="c3">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="polyform.html#K17" title="POLYFORM:func.17">-chain-space</a> <font color="Maroon" title="c1">p</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="polyform.html#K8" title="POLYFORM:func.8">-polytopes</a> <font color="Maroon" title="c1">p</font>;<br/>







<b>set </b><font color="Maroon" title="c5">b</font> =  <a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c3">c</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="polyform.html#K8" title="POLYFORM:func.8">-polytopes</a> <font color="Maroon" title="c1">p</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="polyform.html#K8" title="POLYFORM:func.8">-polytopes</a> <font color="Maroon" title="c1">p</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1"><b>suppose </b></a><a NAME="E1:68_1_1"/><i><font color="Green" title="E56">A1</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="polyform.html#K8" title="POLYFORM:func.8">-polytopes</a> <font color="Maroon" title="c1">p</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:68_1_1"/><b>then </b><i><font color="Green" title="E57">A2</font></i>: 
 <a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="polyform.html#K17" title="POLYFORM:func.17">-chain-space</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="polyform.html#D17" target="_self" title="POLYFORM:def.17">Def17</a></i>;<br/><b>set </b><font color="Maroon" title="c6">iscx</font> =  <a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">c</font>;<br/><a NAME="E3:68_1_1"/>
 <a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 <b>by </b><i><a class="txt" href="polyform.html#E1:68_1_1"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="polyform.html#D16" target="_self" title="POLYFORM:def.16">Def16</a></i>;<br/><a NAME="E4:68_1_1"/><b>then </b>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 <b>by </b><i><a class="ref" href="rlvect_1.html#T60" title="RLVECT_1:th.60">RLVECT_1:60</a></i>;<br/><b>hence </b><a NAME="E5:68_1_1"/>
<span class="p1">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="polyform.html#E2:68_1_1"><i><font color="Green" title="E57">A2</font></i></a>, <a class="ref" href="bspace.html#T14" title="BSPACE:th.14">BSPACE:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_2"><b>suppose </b></a><a NAME="E1:68_1_2"/><i><font color="Green" title="E56">A3</font></i>: 
not <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="polyform.html#K8" title="POLYFORM:func.8">-polytopes</a> <font color="Maroon" title="c1">p</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:68_1_2"/><b>then </b><i><font color="Green" title="E57">A4</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c3">c</font> iff  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>  )
 <b>by </b><i><a class="ref" href="polyform.html#D17" target="_self" title="POLYFORM:def.17">Def17</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_2_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c3">c</font> or  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c3">c</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_2_1_1"><b>suppose </b></a><a NAME="E1:68_1_2_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c3">c</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:68_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="polyform.html#E2:68_1_2"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="bspace.html#D3" title="BSPACE:def.3">BSPACE:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_2_1_2"><b>suppose </b></a><a NAME="E1:68_1_2_1_2"/><i><font color="Green" title="E58">A5</font></i>: 
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c3">c</font>
 ;<br/><div class="add"><a NAME="E2:68_1_2_1_2"/><b>then </b>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 <b>by </b><i><a class="txt" href="polyform.html#E1:68_1_2"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="polyform.html#D17" target="_self" title="POLYFORM:def.17">Def17</a></i>;<br/><a NAME="E3:68_1_2_1_2"/><b>then </b>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 <b>by </b><i><a class="ref" href="bspace.html#T8" title="BSPACE:th.8">BSPACE:8</a></i>;<br/><b>hence </b><a NAME="E4:68_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="polyform.html#E1:68_1_2_1_2"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="bspace.html#D3" title="BSPACE:def.3">BSPACE:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

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