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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:51"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
( 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>) )
 <span class="kw">and </span><br/><a NAME="E2:51"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
( 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>) )
 <span class="kw">and </span><br/><a NAME="E3:51"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">set </span><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="E4:51"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyform.html#T33" target="_self" title="POLYFORM:th.33">Th34</a></span>;<br/>
<a NAME="E5:51"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E6:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A5</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyform.html#T27" target="_self" title="POLYFORM:th.27">Th28</a></span>;<br/>
<a NAME="E7:51"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E8:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A6</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyform.html#T27" target="_self" title="POLYFORM:th.27">Th28</a></span>;<br/>
<a NAME="E9:51"/>
( <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> &amp; <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="polyform.html#D12" target="_self" title="POLYFORM:def.12">Def12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:51"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E8:51"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E6:51"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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