<?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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/> for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> <br/> for <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="polyform.html#K17" title="POLYFORM:func.17">-chain-space</a> <font color="Maroon" title="c1">p</font></span>)</span> holds   <a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Olive" title="b3">c</font></span>)</span></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> <br/> for <font color="Olive" title="b2">c</font> being  <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> holds   <a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b2">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Olive" title="b2">c</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">c</font> being  <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> holds   <a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Olive" title="b1">c</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c4">c</font></span>)</span></span><br/>







<b>set </b><font color="Maroon" title="c5">lsm</font> = <font color="Maroon" title="c3">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c4">c</font>;<br/>
<b>set </b><font color="Maroon" title="c6">l</font> =  <a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c4">c</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">rb</font> =  <a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c4">c</font>;<br/>
<b>set </b><font color="Maroon" title="c8">r</font> = <font color="Maroon" title="c3">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c4">c</font></span>)</span>;<br/>
<a NAME="E1:71"/>
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">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c4">c</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"><font color="Maroon" title="c3">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c4">c</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="71_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c9">x</font></span><br/>

<a NAME="E1:71_1"/><i><font color="Green" title="E59">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">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c9">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="c9">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c4">c</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/>
<a NAME="E2:71_1"/><i><font color="Green" title="E60">A2</font></i>: 
<span class="p1">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c9">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="c9">x</font>,<font color="Maroon" title="c4">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/>
<b>set </b><font color="Maroon" title="c10">b</font> = <span class="p1">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c9">x</font>;<br/>
<a NAME="E3:71_1"/><i><font color="Green" title="E61">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</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">c</font></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c9">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polyform.html#T42" target="_self" title="POLYFORM:th.42">Th42</a></i>;<br/>
<a NAME="E4:71_1"/>
 <a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c9">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c4">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polyform.html#T43" target="_self" title="POLYFORM:th.43">Th43</a></i>;<br/>
<b>hence </b><a NAME="E5:71_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">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="polyform.html#E1:71_1"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="polyform.html#E2:71_1"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="polyform.html#E3:71_1"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="fvsum_1.html#T92" title="FVSUM_1:th.92">FVSUM_1:92</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:71"/>
 <a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><a href="polyform.html#K20" title="POLYFORM:func.20">Boundary</a> <font color="Maroon" title="c4">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="polyform.html#T44" target="_self" title="POLYFORM:th.44">Th44</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
