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<b>let </b><font color="Maroon" title="c1">p</font> be    <a href="polyform.html#M1" title="POLYFORM:mode.1">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>assume </b><a NAME="E1:48"/><i><font color="Green" title="E40">A1</font></i>: 
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font> &amp; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">k</font> <a href="polyform.html#K8" title="POLYFORM:func.8">-polytopes</a> <font color="Maroon" title="c1">p</font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1"/>
( <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 or <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font> or (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">k</font> &amp; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font> ) )
 </span><b>by </b><i><a class="txt" href="polyform.html#E1:48"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1"><b>suppose </b></a><a NAME="E1:48_1_1"/><i><font color="Green" title="E41">A2</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 ;<br/><div class="add"><a NAME="E2:48_1_1"/><i><font color="Green" title="E42">A3</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="polyform.html#K8" title="POLYFORM:func.8">-polytopes</a> <font color="Maroon" title="c1">p</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"><a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="polyform.html#E1:48_1_1"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="polyform.html#D3" target="_self" title="POLYFORM:def.3">Def3</a></i>;<br/><a NAME="E3:48_1_1"/><i><font color="Green" title="E43">A4</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 <b>by </b><i><a class="txt" href="polyform.html#E2:48_1_1"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">n</font> = 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><a NAME="E5:48_1_1"/><i><font color="Green" title="E44">A5</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="polyform.html#K10" title="POLYFORM:func.10">-polytope-seq</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polyform.html#K3" title="POLYFORM:func.3">&lt;*</a><span class="default"><a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> </span><a href="polyform.html#K3" title="POLYFORM:func.3">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polyform.html#E1:48_1_1"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="polyform.html#D5" target="_self" title="POLYFORM:def.5">Def5</a></i>;<br/><a NAME="E6:48_1_1"/><i><font color="Green" title="E45">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="polyform.html#K10" title="POLYFORM:func.10">-polytope-seq</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 <b>by </b><i><a class="txt" href="polyform.html#E5:48_1_1"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/><a NAME="E7:48_1_1"/><i><font color="Green" title="E46">A7</font></i>: 
<font color="Maroon" title="c4">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>
 <b>by </b><i><a class="txt" href="polyform.html#E2:48_1_1"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a></i>;<br/><b>take </b>
<font color="Maroon" title="c4">n</font>
;<br/><b>thus </b><a NAME="E8:48_1_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c4">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> )
 <b>by </b><i><a class="txt" href="polyform.html#E1:48"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="polyform.html#E3:48_1_1"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="polyform.html#E6:48_1_1"><i><font color="Green" title="E45">A6</font></i></a>, <a class="txt" href="polyform.html#E7:48_1_1"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="polyform.html#D10" target="_self" title="POLYFORM:def.10">Def10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2"><b>suppose </b></a><a NAME="E1:48_1_2"/><i><font color="Green" title="E41">A8</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font>
 ;<br/><div class="add"><a NAME="E2:48_1_2"/><i><font color="Green" title="E42">A9</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="polyform.html#K8" title="POLYFORM:func.8">-polytopes</a> <font color="Maroon" title="c1">p</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>
 <b>by </b><i><a class="txt" href="polyform.html#E1:48_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="ref" href="polyform.html#D3" target="_self" title="POLYFORM:def.3">Def3</a></i>;<br/><a NAME="E3:48_1_2"/><i><font color="Green" title="E43">A10</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="polyform.html#E2:48_1_2"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">n</font> = 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><a NAME="E5:48_1_2"/><i><font color="Green" title="E44">A11</font></i>: 
 <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="polyform.html#E2:48_1_2"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a></i>;<br/><a NAME="E6:48_1_2"/><i><font color="Green" title="E45">A12</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="polyform.html#K10" title="POLYFORM:func.10">-polytope-seq</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polyform.html#E1:48_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="ref" href="polyform.html#D5" target="_self" title="POLYFORM:def.5">Def5</a></i>;<br/><a NAME="E7:48_1_2"/><i><font color="Green" title="E46">A13</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="polyform.html#K10" title="POLYFORM:func.10">-polytope-seq</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="polyform.html#E6:48_1_2"><i><font color="Green" title="E45">A12</font></i></a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/><b>take </b>
<font color="Maroon" title="c4">n</font>
;<br/><b>thus </b><a NAME="E8:48_1_2"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c4">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> )
 <b>by </b><i><a class="txt" href="polyform.html#E1:48"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="polyform.html#E3:48_1_2"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="polyform.html#E5:48_1_2"><i><font color="Green" title="E44">A11</font></i></a>, <a class="txt" href="polyform.html#E7:48_1_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="ref" href="polyform.html#D10" target="_self" title="POLYFORM:def.10">Def10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_3"><b>suppose </b></a><a NAME="E1:48_1_3"/><i><font color="Green" title="E41">A14</font></i>: 
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">k</font> &amp; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c4">F</font> being  <a href="matrlin.html#V1" title="MATRLIN:attr.1">FinSequence-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>, <font color="Maroon" title="c5">I</font> being  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E3:48_1_3"/><i><font color="Green" title="E42">A15</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>and </b><br/><a NAME="E4:48_1_3"/><i><font color="Green" title="E43">A16</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font> holds <br/><font color="Maroon" title="c5">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is   <a href="polyform.html#NM1" title="POLYFORM:NM.1">incidence-matrix</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:48_1_3"/><i><font color="Green" title="E44">A17</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font> holds <br/>( not <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>and </b><br/><a NAME="E6:48_1_3"/><i><font color="Green" title="E45">A18</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">I</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:48_1_3"/><i><font color="Green" title="E46">A19</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="polyform.html#K8" title="POLYFORM:func.8">-polytopes</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polyform.html#E1:48_1_3"><i><font color="Green" title="E41">A14</font></i></a>, <a class="ref" href="polyform.html#D3" target="_self" title="POLYFORM:def.3">Def3</a></i>;<br/><a NAME="E8:48_1_3"/><i><font color="Green" title="E47">A20</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="polyform.html#K10" title="POLYFORM:func.10">-polytope-seq</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="polyform.html#E1:48_1_3"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="polyform.html#E3:48_1_3"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="polyform.html#E4:48_1_3"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="polyform.html#E5:48_1_3"><i><font color="Green" title="E44">A17</font></i></a>, <a class="txt" href="polyform.html#E6:48_1_3"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="polyform.html#D5" target="_self" title="POLYFORM:def.5">Def5</a></i>;<br/><a NAME="E9:48_1_3"/><i><font color="Green" title="E48">A21</font></i>: 
 <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polyform.html#E8:48_1_3"><i><font color="Green" title="E47">A20</font></i></a>, <a class="ref" href="polyform.html#T29" target="_self" title="POLYFORM:th.29">Th29</a></i>;<br/><a NAME="E10:48_1_3"/><i><font color="Green" title="E49">A22</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="polyform.html#E3:48_1_3"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="polyform.html#E4:48_1_3"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="polyform.html#E5:48_1_3"><i><font color="Green" title="E44">A17</font></i></a>, <a class="txt" href="polyform.html#E6:48_1_3"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="polyform.html#D2" target="_self" title="POLYFORM:def.2">Def2</a></i>;<br/><a NAME="E11:48_1_3"/><i><font color="Green" title="E50">A23</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 <b>by </b><i><a class="txt" href="polyform.html#E1:48_1_3"><i><font color="Green" title="E41">A14</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E12:48_1_3"/><i><font color="Green" title="E51">A24</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="polyform.html#E1:48_1_3"><i><font color="Green" title="E41">A14</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><a NAME="E13:48_1_3"/><i><font color="Green" title="E52">A25</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 <b>by </b><i><a class="txt" href="polyform.html#E11:48_1_3"><i><font color="Green" title="E50">A23</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">k'</font> = <font color="Maroon" title="c2">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 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="txt" href="polyform.html#E11:48_1_3"><i><font color="Green" title="E50">A23</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E15:48_1_3"/><i><font color="Green" title="E53">A26</font></i>: 
not <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="polyform.html#E5:48_1_3"><i><font color="Green" title="E44">A17</font></i></a>, <a class="txt" href="polyform.html#E10:48_1_3"><i><font color="Green" title="E49">A22</font></i></a>, <a class="txt" href="polyform.html#E12:48_1_3"><i><font color="Green" title="E51">A24</font></i></a>, <a class="txt" href="polyform.html#E13:48_1_3"><i><font color="Green" title="E52">A25</font></i></a></i>;<br/><a NAME="E16:48_1_3"/><i><font color="Green" title="E54">A27</font></i>: 
not  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k'</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="polyform.html#E15:48_1_3"><i><font color="Green" title="E53">A26</font></i></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E18:48_1_3"/><i><font color="Green" title="E55">A28</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k'</font></span>)</span>
 <b>and </b><br/><a NAME="E19:48_1_3"/><i><font color="Green" title="E56">A29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="polyform.html#E7:48_1_3"><i><font color="Green" title="E46">A19</font></i></a>, <a class="txt" href="polyform.html#E16:48_1_3"><i><font color="Green" title="E54">A27</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">Fk'</font> = <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k'</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/><a NAME="E21:48_1_3"/><i><font color="Green" title="E57">A30</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">Fk'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">Fk'</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_3_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c9">a</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E2:48_1_3_1"/><i><font color="Green" title="E58">A31</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">Fk'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/><b>thus </b><a NAME="E3:48_1_3_1"/>
<font color="Maroon" title="c7">m</font> is    <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="txt" href="polyform.html#E18:48_1_3"><i><font color="Green" title="E55">A28</font></i></a>, <a class="txt" href="polyform.html#E2:48_1_3_1"><i><font color="Green" title="E58">A31</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c9">m</font> = <font color="Maroon" title="c7">m</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>  ;<br/><a NAME="E24:48_1_3"/><i><font color="Green" title="E58">A32</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">Fk'</font> )
 <b>by </b><i><a class="txt" href="polyform.html#E18:48_1_3"><i><font color="Green" title="E55">A28</font></i></a>, <a class="txt" href="polyform.html#E21:48_1_3"><i><font color="Green" title="E57">A30</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>take </b>
<font color="Maroon" title="c9">m</font>
;<br/><b>thus </b><a NAME="E25:48_1_3"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">m</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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c9">m</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> )
 <b>by </b><i><a class="txt" href="polyform.html#E1:48_1_3"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="polyform.html#E8:48_1_3"><i><font color="Green" title="E47">A20</font></i></a>, <a class="txt" href="polyform.html#E9:48_1_3"><i><font color="Green" title="E48">A21</font></i></a>, <a class="txt" href="polyform.html#E19:48_1_3"><i><font color="Green" title="E56">A29</font></i></a>, <a class="txt" href="polyform.html#E24:48_1_3"><i><font color="Green" title="E58">A32</font></i></a>, <a class="ref" href="polyform.html#D10" target="_self" title="POLYFORM:def.10">Def10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
