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<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">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font></span>)</span> <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/> 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"><span class="p2">(<span class="default"><a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="polyform.html#K17" title="POLYFORM:func.17">-chain-space</a> <font color="Maroon" title="c1">p</font></span>)</span>  st <font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/> <a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> </span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><b>let </b><font color="Maroon" title="c2">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font></span>)</span> <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"> 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"><span class="p2">(<span class="default"><a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="polyform.html#K17" title="POLYFORM:func.17">-chain-space</a> <font color="Maroon" title="c1">p</font></span>)</span>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/> <a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> </span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><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"><span class="p2">(<span class="default"><a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font></span>)</span> <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"> ( <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  <a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> </span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/>





<b>assume </b><a NAME="E1:111"/><i><font color="Green" title="E91">A1</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> </span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/>

<b>set </b><font color="Maroon" title="c4">iseq</font> =  <a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">c</font>;<br/>
<a NAME="E2:111"/>
 <a href="polyform.html#K11" title="POLYFORM:func.11">num-polytopes</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="polyform.html#T32" target="_self" title="POLYFORM:th.32">Th32</a></i>;<br/>
<a NAME="E3:111"/><b>then </b><i><font color="Green" title="E92">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="polyform.html#D16" target="_self" title="POLYFORM:def.16">Def16</a></i>;<br/>
<a NAME="E4:111"/>
<span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c5">egy</font> = 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<a NAME="E2:111_1"/><i><font color="Green" title="E93">A3</font></i>: 
<font color="Maroon" title="c5">egy</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="polyform.html#K11" title="POLYFORM:func.11">num-polytopes</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polyform.html#T32" target="_self" title="POLYFORM:th.32">Th32</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">z</font> = <font color="Maroon" title="c5">egy</font> <a href="polyform.html#K15" title="POLYFORM:func.15">-th-polytope</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font></span>)</span>;<br/>
<a NAME="E3:111_1"/><i><font color="Green" title="E94">A4</font></i>: 
<span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">egy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">egy</font> <a href="polyform.html#K15" title="POLYFORM:func.15">-th-polytope</a> <font color="Maroon" title="c1">p</font>,<span class="p3">(<span class="default"><a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="polyform.html#K16" title="POLYFORM:func.16">incidence-value</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">egy</font> <a href="polyform.html#K15" title="POLYFORM:func.15">-th-polytope</a> <font color="Maroon" title="c1">p</font>,<span class="p3">(<span class="default"><a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polyform.html#E2:111_1"><i><font color="Green" title="E93">A3</font></i></a>, <a class="ref" href="polyform.html#D16" target="_self" title="POLYFORM:def.16">Def16</a></i>;<br/>
<a NAME="E4:111_1"/><i><font color="Green" title="E95">A5</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">egy</font> <a href="polyform.html#K15" title="POLYFORM:func.15">-th-polytope</a> <font color="Maroon" title="c1">p</font>,<span class="p2">(<span class="default"><a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font></span>)</span></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="txt" href="polyform.html#E1:111"><i><font color="Green" title="E91">A1</font></i></a>, <a class="ref" href="polyform.html#T73" target="_self" title="POLYFORM:th.73">Th73</a></i>;<br/>
<a NAME="E5:111_1"/>
 <a href="polyform.html#K16" title="POLYFORM:func.16">incidence-value</a> <font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">egy</font> <a href="polyform.html#K15" title="POLYFORM:func.15">-th-polytope</a> <font color="Maroon" title="c1">p</font>,<span class="p2">(<span class="default"><a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font></span>)</span></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#T72" target="_self" title="POLYFORM:th.72">Th72</a>, <a class="ref" href="polyform.html#T74" target="_self" title="POLYFORM:th.74">Th74</a></i>;<br/>
<b>hence </b><a NAME="E6:111_1"/>
<span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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="txt" href="polyform.html#E3:111_1"><i><font color="Green" title="E94">A4</font></i></a>, <a class="txt" href="polyform.html#E4:111_1"><i><font color="Green" title="E95">A5</font></i></a>, <a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</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="E5:111"/>
 <a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> </span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polyform.html#E3:111"><i><font color="Green" title="E92">A2</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</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>
