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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">KX</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">P</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/>  for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<span class="p1">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b4">k</font></span>)</span>,<font color="Olive" title="b2">P</font>,<font color="Olive" title="b1">KX</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Olive" title="b3">n</font>,<font color="Olive" title="b2">P</font>,<span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Olive" title="b4">k</font>,<font color="Olive" title="b2">P</font>,<font color="Olive" title="b1">KX</font>)</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">KX</font> be    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font>; <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">P</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/>  for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">k</font></span>)</span>,<font color="Olive" title="b1">P</font>,<font color="Maroon" title="c2">KX</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Olive" title="b2">n</font>,<font color="Olive" title="b1">P</font>,<span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Olive" title="b3">k</font>,<font color="Olive" title="b1">P</font>,<font color="Maroon" title="c2">KX</font>)</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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">n</font>, <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">k</font></span>)</span>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c3">P</font>,<span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Olive" title="b2">k</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>)</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">n</font>, <font color="Maroon" title="c5">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c3">P</font>,<span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>)</span>)</span>)</span><br/>







<span class="kw">consider </span><font color="Maroon" title="c6">Fn</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:142"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
<font color="Maroon" title="c6">Fn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>)
 <span class="kw">and </span><br/><a NAME="E3:142"/><span class="lab"><font color="Green" title="E59">A2</font></span>: 
<font color="Maroon" title="c6">Fn</font> <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="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c3">P</font>,<span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>)</span>)</span>)
 <span class="kw">and </span><br/><a NAME="E4:142"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Fn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">and </span><br/><a NAME="E5:142"/><span class="lab"><font color="Green" title="E60">A3</font></span>: 
 for <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b2">KX1</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">KX1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Fn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c6">Fn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Olive" title="b2">KX1</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D21" target="_self" title="SIMPLEX0:def.21">Def21</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">Fnm</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E7:142"/><span class="lab"><font color="Green" title="E61">A4</font></span>: 
<font color="Maroon" title="c7">Fnm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">KX</font>
 <span class="kw">and </span><br/><a NAME="E8:142"/><span class="lab"><font color="Green" title="E62">A5</font></span>: 
<font color="Maroon" title="c7">Fnm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>)
 <span class="kw">and </span><br/><a NAME="E9:142"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Fnm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">and </span><br/><a NAME="E10:142"/><span class="lab"><font color="Green" title="E63">A6</font></span>: 
 for <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b2">KX1</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">KX1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Fnm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c7">Fnm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Olive" title="b2">KX1</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D21" target="_self" title="SIMPLEX0:def.21">Def21</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> implies ( <font color="Maroon" title="c6">Fn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Fnm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">k</font></span>)</span> &amp;  ex <font color="Olive" title="b1">SX</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font> st <font color="Maroon" title="c6">Fn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">SX</font> ) );<br/>
<a NAME="E11:142"/><span class="lab"><font color="Green" title="E64">A7</font></span>: 
 for <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="142_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">m</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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">m</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:142_1"/><span class="lab"><font color="Green" title="E65">A8</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">m</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">S<sub>1</sub></font>[<font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">assume </span><a NAME="E2:142_1"/><span class="lab"><font color="Green" title="E66">A9</font></span>: 
<font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</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="c6">Fn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Fnm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">k</font></span>)</span> &amp;  ex <font color="Olive" title="b1">SX</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font> st <font color="Maroon" title="c6">Fn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">SX</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">SX</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E4:142_1"/><span class="lab"><font color="Green" title="E67">A10</font></span>: 
<font color="Maroon" title="c6">Fn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">SX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:142_1"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E5:142_1"/>
 <a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c9">SX</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Fnm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:142"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E1:142_1"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="txt" href="#E2:142_1"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="txt" href="#E4:142_1"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:142_1"/>
( <font color="Maroon" title="c6">Fn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Fnm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">k</font></span>)</span> &amp;  ex <font color="Olive" title="b1">SX</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font> st <font color="Maroon" title="c6">Fn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">SX</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:142"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="txt" href="#E4:142_1"><span class="lab"><font color="Green" title="E67">A10</font></span></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/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c8">Fm</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E13:142"/><span class="lab"><font color="Green" title="E65">A11</font></span>: 
<font color="Maroon" title="c8">Fm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">KX</font>
 <span class="kw">and </span><br/><a NAME="E14:142"/><span class="lab"><font color="Green" title="E66">A12</font></span>: 
<font color="Maroon" title="c8">Fm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>)
 <span class="kw">and </span><br/><a NAME="E15:142"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">Fm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">and </span><br/><a NAME="E16:142"/><span class="lab"><font color="Green" title="E67">A13</font></span>: 
 for <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b2">KX1</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">KX1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Fm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c8">Fm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Olive" title="b2">KX1</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D21" target="_self" title="SIMPLEX0:def.21">Def21</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> implies ( <font color="Maroon" title="c8">Fm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Fnm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp;  ex <font color="Olive" title="b1">SX</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font> st <font color="Maroon" title="c8">Fm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">SX</font> ) );<br/>
<a NAME="E17:142"/><span class="lab"><font color="Green" title="E68">A14</font></span>: 
 for <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="142_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">m</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"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">m</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:142_2"/><span class="lab"><font color="Green" title="E69">A15</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">m</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">S<sub>2</sub></font>[<font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">assume </span><a NAME="E2:142_2"/><span class="lab"><font color="Green" title="E70">A16</font></span>: 
<font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">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="c8">Fm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Fnm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp;  ex <font color="Olive" title="b1">SX</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font> st <font color="Maroon" title="c8">Fm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">SX</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">SX</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E4:142_2"/><span class="lab"><font color="Green" title="E71">A17</font></span>: 
<font color="Maroon" title="c8">Fm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">SX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:142_2"><span class="lab"><font color="Green" title="E69">A15</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E5:142_2"/>
 <a href="simplex0.html#K10" title="SIMPLEX0:func.10">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c10">SX</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Fnm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:142"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E1:142_2"><span class="lab"><font color="Green" title="E69">A15</font></span></a>, <a class="txt" href="#E2:142_2"><span class="lab"><font color="Green" title="E70">A16</font></span></a>, <a class="txt" href="#E4:142_2"><span class="lab"><font color="Green" title="E71">A17</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:142_2"/>
( <font color="Maroon" title="c8">Fm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Fnm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp;  ex <font color="Olive" title="b1">SX</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font> st <font color="Maroon" title="c8">Fm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">SX</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:142"><span class="lab"><font color="Green" title="E67">A13</font></span></a>, <a class="txt" href="#E4:142_2"><span class="lab"><font color="Green" title="E71">A17</font></span></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/>


</div><span class="kw">end;</span></div>
<a NAME="E18:142"/><span class="lab"><font color="Green" title="E69">A18</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:142"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E13:142"><span class="lab"><font color="Green" title="E65">A11</font></span></a></span>;<br/>
<a NAME="E19:142"/>
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E18:142"><span class="lab"><font color="Green" title="E69">A18</font></span></a>, <a class="txt" href="#E17:142"><span class="lab"><font color="Green" title="E68">A14</font></span></a>);<br/></span>
<a NAME="E20:142"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A19</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:142"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E14:142"><span class="lab"><font color="Green" title="E66">A12</font></span></a></span>;<br/>
<a NAME="E21:142"/>
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E20:142"><span class="lab"><font color="Green" title="E70">A19</font></span></a>, <a class="txt" href="#E11:142"><span class="lab"><font color="Green" title="E64">A7</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E22:142"/>
 <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c3">P</font>,<span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:142"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E8:142"><span class="lab"><font color="Green" title="E62">A5</font></span></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/>


</div>
