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<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>  st  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">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> holds <br/>for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">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="b3">n</font> &amp; <font color="Olive" title="b3">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="Olive" title="b1">k</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">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="Olive" title="b1">k</font> &amp; <font color="Olive" title="b3">n</font> <a href="polyform.html#K15" title="POLYFORM:func.15">-th-polytope</a> <font color="Maroon" title="c1">p</font>,<font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> <a href="polyform.html#K15" title="POLYFORM:func.15">-th-polytope</a> <font color="Maroon" title="c1">p</font>,<font color="Olive" title="b1">k</font> holds <br/><font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">n</font></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"> (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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> implies for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">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="b2">n</font> &amp; <font color="Olive" title="b2">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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">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> &amp; <font color="Olive" title="b2">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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> holds <br/><font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n</font> )</span><br/>



<b>assume </b><a NAME="E1:52"/><i><font color="Green" title="E43">A1</font></i>: 
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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> )
 ; <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">m</font>, <font color="Olive" title="b2">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="b2">n</font> &amp; <font color="Olive" title="b2">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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">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> &amp; <font color="Olive" title="b2">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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> holds <br/><font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n</font></span><br/>

<b>let </b><font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c3">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> &amp; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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> implies <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:52"/><i><font color="Green" title="E44">A2</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 <b>and </b><br/><a NAME="E3:52"/><i><font color="Green" title="E45">A3</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>and </b><br/><a NAME="E4:52"/><i><font color="Green" title="E46">A4</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font>
 <b>and </b><br/><a NAME="E5:52"/><i><font color="Green" title="E47">A5</font></i>: 
<font color="Maroon" title="c3">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>and </b><br/><a NAME="E6:52"/><i><font color="Green" title="E48">A6</font></i>: 
<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font></span><br/>

<b>set </b><font color="Maroon" title="c5">s</font> = <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>;<br/>
<a NAME="E7:52"/><i><font color="Green" title="E49">A7</font></i>: 
<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>
 
<b>by </b><i><a class="txt" href="#E1:52"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="#E2:52"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="#E3:52"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="polyform.html#D12" target="_self" title="POLYFORM:def.12">Def12</a></i>;<br/>
<a NAME="E8:52"/><i><font color="Green" title="E50">A8</font></i>: 
<font color="Maroon" title="c3">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c3">m</font>
 
<b>by </b><i><a class="txt" href="#E1:52"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="#E4:52"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="#E5:52"><i><font color="Green" title="E47">A5</font></i></a>, <a class="ref" href="polyform.html#D12" target="_self" title="POLYFORM:def.12">Def12</a></i>;<br/>
<a NAME="E9:52"/>
<font color="Maroon" title="c4">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="#E2:52"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="#E3:52"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:52"/><b>then </b><i><font color="Green" title="E51">A9</font></i>: 
<font color="Maroon" title="c4">n</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="c2">k</font> <a href="polyform.html#K10" title="POLYFORM:func.10">-polytope-seq</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polyform.html#T28" target="_self" title="POLYFORM:th.28">Th28</a></i>;<br/>
<a NAME="E11:52"/>
<font color="Maroon" title="c3">m</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="#E4:52"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="#E5:52"><i><font color="Green" title="E47">A5</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E12:52"/><b>then </b><i><font color="Green" title="E52">A10</font></i>: 
<font color="Maroon" title="c3">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="c2">k</font> <a href="polyform.html#K10" title="POLYFORM:func.10">-polytope-seq</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polyform.html#T28" target="_self" title="POLYFORM:th.28">Th28</a></i>;<br/>
<a NAME="E13:52"/>
<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> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="polyform.html#T34" target="_self" title="POLYFORM:th.34">Th34</a></i>;<br/>
<b>hence </b><a NAME="E14:52"/>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="#E6:52"><i><font color="Green" title="E48">A6</font></i></a>, <a class="txt" href="#E7:52"><i><font color="Green" title="E49">A7</font></i></a>, <a class="txt" href="#E8:52"><i><font color="Green" title="E50">A8</font></i></a>, <a class="txt" href="#E10:52"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="#E12:52"><i><font color="Green" title="E52">A10</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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