<?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>;<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>, <font color="Maroon" title="c4">d</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>set </b><font color="Maroon" title="c5">bc</font> =  <a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c3">c</font>;<br/>
<b>set </b><font color="Maroon" title="c6">bd</font> =  <a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c4">d</font>;<br/>
<b>set </b><font color="Maroon" title="c7">s</font> = <font color="Maroon" title="c3">c</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">d</font>;<br/>
<b>set </b><font color="Maroon" title="c8">l</font> =  <a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">d</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">r</font> = <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="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c4">d</font></span>)</span>;<br/>
<a NAME="E1:70"/>
for <font color="Olive" title="b1">x</font> being   <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> holds  <span class="p1">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">d</font></span>)</span></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Olive" title="b1">x</font> <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#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c4">d</font></span>)</span></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">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/>

<a NAME="E1:70_1"/><i><font color="Green" title="E58">A1</font></i>: 
<span class="p1">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">d</font></span>)</span></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c10">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="c10">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">d</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="polyform.html#T46" target="_self" title="POLYFORM:th.46">Th46</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">a</font> = <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="c10">x</font>;<br/>
<b>set </b><font color="Maroon" title="c12">b</font> = <span class="p1">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c10">x</font>;<br/>
<a NAME="E2:70_1"/><i><font color="Green" title="E59">A2</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<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="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c4">d</font></span>)</span></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c10">x</font> <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#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="c10">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c10">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polyform.html#T38" target="_self" title="POLYFORM:th.38">Th38</a></i>;<br/>
<a NAME="E3:70_1"/><i><font color="Green" title="E60">A3</font></i>: 
<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="c10">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="c10">x</font>,<font color="Maroon" title="c3">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polyform.html#T46" target="_self" title="POLYFORM:th.46">Th46</a></i>;<br/>
<a NAME="E4:70_1"/>
<span class="p1">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c10">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="c10">x</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polyform.html#T46" target="_self" title="POLYFORM:th.46">Th46</a></i>;<br/>
<b>hence </b><a NAME="E5:70_1"/>
<span class="p1">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">d</font></span>)</span></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c10">x</font> <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#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c4">d</font></span>)</span></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="polyform.html#E1:70_1"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="polyform.html#E2:70_1"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="polyform.html#E3:70_1"><i><font color="Green" title="E60">A3</font></i></a>, <a class="ref" href="polyform.html#T41" target="_self" title="POLYFORM:th.41">Th41</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:70"/>
 <a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c4">d</font></span>)</span>
 <b>by </b><i><a class="ref" href="polyform.html#T44" target="_self" title="POLYFORM:th.44">Th44</a></i>;<br/>


</div>
