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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="real_1.html#K5" title="REAL_1:func.5">-</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="55_1_1_1_1"><b>suppose </b></a><a NAME="E1:55_1_1_1_1"/><i><font color="Green" title="E44">A1</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</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"><b>set </b><font color="Maroon" title="c5">s</font> =  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> ;<br/><a NAME="E2:55_1_1_1_1"/><i><font color="Green" title="E45">A2</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">s</font> =  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>  <b>by </b><i><a class="txt" href="polyform.html#E2:55_1_1_1_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><b>take </b>
<font color="Maroon" title="c6">s</font>
;<br/><b>thus </b><a NAME="E4:55_1_1_1_1"/>
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</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> implies <font color="Maroon" title="c6">s</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> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  ) &amp; ( not <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</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> implies (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyform.html#K11" title="POLYFORM:func.11">num-polytopes</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">k</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</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>,<font color="Maroon" title="c2">k</font> holds <br/><font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="polyform.html#K15" title="POLYFORM:func.15">-th-polytope</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polyform.html#K16" title="POLYFORM:func.16">incidence-value</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="polyform.html#K15" title="POLYFORM:func.15">-th-polytope</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span> ) ) ) )
 <b>by </b><i><a class="txt" href="polyform.html#E1:55_1_1_1_1"><i><font color="Green" title="E44">A1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1_2"><b>suppose </b></a><a NAME="E1:55_1_1_1_2"/><i><font color="Green" title="E44">A3</font></i>: 
not <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</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"><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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>  = <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <span class="p2">(<span class="default">$1 <a href="polyform.html#K15" title="POLYFORM:func.15">-th-polytope</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polyform.html#K16" title="POLYFORM:func.16">incidence-value</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default">$1 <a href="polyform.html#K15" title="POLYFORM:func.15">-th-polytope</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c5">s</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> <b> such that </b><br/><a NAME="E3:55_1_1_1_2"/><i><font color="Green" title="E45">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyform.html#K11" title="POLYFORM:func.11">num-polytopes</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">k</font>
 <b>and </b><br/><a NAME="E4:55_1_1_1_2"/><i><font color="Green" title="E46">A5</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,<font color="Maroon" title="c2">k</font></span>)</span> holds <br/><font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i><a NAME="E5:55_1_1_1_2"/><i><font color="Green" title="E47">A6</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</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>,<font color="Maroon" title="c2">k</font> holds <br/><font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="polyform.html#K15" title="POLYFORM:func.15">-th-polytope</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polyform.html#K16" title="POLYFORM:func.16">incidence-value</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="polyform.html#K15" title="POLYFORM:func.15">-th-polytope</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:55_1_1_1_2_1"/><i><font color="Green" title="E48">A7</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 <b>and </b><br/><a NAME="E2:55_1_1_1_2_1"/><i><font color="Green" title="E49">A8</font></i>: 
<font color="Maroon" title="c6">n</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>,<font color="Maroon" title="c2">k</font>
 ;<br/>

<a NAME="E3:55_1_1_1_2_1"/><i><font color="Green" title="E50">A9</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,<font color="Maroon" title="c2">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polyform.html#E1:55_1_1_1_2_1"><i><font color="Green" title="E48">A7</font></i></a>, <a class="txt" href="polyform.html#E2:55_1_1_1_2_1"><i><font color="Green" title="E49">A8</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">n</font> = <font color="Maroon" title="c6">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>thus </b><a NAME="E5:55_1_1_1_2_1"/>
<font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="polyform.html#K15" title="POLYFORM:func.15">-th-polytope</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polyform.html#K16" title="POLYFORM:func.16">incidence-value</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="polyform.html#K15" title="POLYFORM:func.15">-th-polytope</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polyform.html#E4:55_1_1_1_2"><i><font color="Green" title="E46">A5</font></i></a>, <a class="txt" href="polyform.html#E3:55_1_1_1_2_1"><i><font color="Green" title="E50">A9</font></i></a></i>;<br/>


</div><b>end;</b></div><b>take </b>
<font color="Maroon" title="c5">s</font>
;<br/><b>thus </b><a NAME="E6:55_1_1_1_2"/>
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</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> implies <font color="Maroon" title="c5">s</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> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  ) &amp; ( not <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</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> implies (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyform.html#K11" title="POLYFORM:func.11">num-polytopes</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">k</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</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>,<font color="Maroon" title="c2">k</font> holds <br/><font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="polyform.html#K15" title="POLYFORM:func.15">-th-polytope</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polyform.html#K16" title="POLYFORM:func.16">incidence-value</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="polyform.html#K15" title="POLYFORM:func.15">-th-polytope</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">k</font></span>)</span></span>)</span> ) ) ) )
 <b>by </b><i><a class="txt" href="polyform.html#E1:55_1_1_1_2"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="polyform.html#E3:55_1_1_1_2"><i><font color="Green" title="E45">A4</font></i></a>, <a class="txt" href="polyform.html#E5:55_1_1_1_2"><i><font color="Green" title="E47">A6</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
