<?xml version="1.0"?>
<div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> $1,0,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">s</font></span>)</span> where <font color="Olive" title="b1">s</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c2">Sk</font> being   <a href="triang_1.html#NM2" title="TRIANG_1:NM.2">SetSequence</a><b> such that </b><br/><a NAME="E2:45_1_1"/><i><font color="Green" title="E14">A1</font></i>: 
for <font color="Olive" title="b1">n</font> being   <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>  holds  <font color="Maroon" title="c2">Sk</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="pboole.html#S5" title="PBOOLE:sch.5">PBOOLE:sch 5</a>();<br/></i>
<i><font color="Green" title="E15">A2</font></i>: <font color="Maroon" title="c2">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 = 
 <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> 0,0,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">s</font></span>)</span> where <font color="Olive" title="b1">s</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 </span>}</span> 
<b>by </b><i><a class="txt" href="triang_1.html#E2:45_1_1"><i><font color="Green" title="E14">A1</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
;<br/>
<div><i><font color="Green" title="E16">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:45_1_1_2"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:45_1_1_2"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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>hence </b><font color="Maroon" title="c2">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> = 
 <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> <font color="Maroon" title="c3">n</font>,0,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">s</font></span>)</span> where <font color="Olive" title="b1">s</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> </span>}</span> 
<b>by </b><i><a class="txt" href="triang_1.html#E2:45_1_1"><i><font color="Green" title="E14">A1</font></i></a></i>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">s</font></span>)</span> where <font color="Olive" title="b1">s</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> </span>}</span> 
<b>by </b><i><a class="txt" href="triang_1.html#E1:45_1_1_2"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E5:45_1_1"/>
<font color="Maroon" title="c2">Sk</font> is <a href="triang_1.html#V1" title="TRIANG_1:attr.1">lower_non-empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="triang_1.html#D4" target="_self" title="TRIANG_1:def.4">TRIANG_1:def 4</a><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>not <font color="Maroon" title="c2">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>;<br/>
<b>assume </b><a NAME="E1:45_1_1_3"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>]
 ;<br/>

<a NAME="E2:45_1_1_3"/><i><font color="Green" title="E18">A6</font></i>: 
for <font color="Olive" title="b1">m</font> being   <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>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> &amp; <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] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">m</font> be    <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/>

<b>assume </b><a NAME="E1:45_1_1_3_1"/>
( <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> &amp; not <font color="Maroon" title="c2">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">g</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:45_1_1_3_1"/><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:45_1_1_3_1"/>
<font color="Maroon" title="c2">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">s</font></span>)</span> where <font color="Olive" title="b1">s</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 </span>}</span> 
 
<b>by </b><i><a class="txt" href="triang_1.html#E4:45_1_1"><i><font color="Green" title="E16">A3</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">s</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E6:45_1_1_3_1"/><i><font color="Green" title="E20">A8</font></i>: 
( <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">s</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="triang_1.html#E3:45_1_1_3_1"><i><font color="Green" title="E19">A7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">s</font>;<br/>
<a NAME="E8:45_1_1_3_1"/><i><font color="Green" title="E21">A9</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">s</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">sx</font> = <font color="Maroon" title="c6">s</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E10:45_1_1_3_1"/>
<font color="Maroon" title="c8">sx</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>;<br/>
<a NAME="E11:45_1_1_3_1"/><b>then </b><i><font color="Green" title="E22">A10</font></i>: 
<font color="Maroon" title="c8">sx</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E12:45_1_1_3_1"/>
not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">sx</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E13:45_1_1_3_1"/><b>then </b><i><font color="Green" title="E23">A11</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c8">sx</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="triang_1.html#E6:45_1_1_3_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="txt" href="triang_1.html#E11:45_1_1_3_1"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_1_3_1_1"/>
( <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c4">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_1_3_1_1_1"/>
<font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:45_1_1_3_1_1_1"/>
not <font color="Maroon" title="c2">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="triang_1.html#E3:45_1_1"><i><font color="Green" title="E15">A2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_1_3_1_1_2"/><i><font color="Green" title="E24">A12</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c9">sx</font> = <font color="Maroon" title="c8">sx</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="txt" href="triang_1.html#E8:45_1_1_3_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="txt" href="triang_1.html#E13:45_1_1_3_1"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="triang_1.html#T7" target="_self" title="TRIANG_1:th.7">Th7</a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/><a NAME="E3:45_1_1_3_1_1_2"/>
 <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c9">sx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">s1</font></span>)</span> where <font color="Olive" title="b1">s1</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="triang_1.html#E13:45_1_1_3_1"><i><font color="Green" title="E23">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E4:45_1_1_3_1_1_2"/>
not <font color="Maroon" title="c2">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="triang_1.html#E4:45_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="triang_1.html#E1:45_1_1_3_1_1_2"><i><font color="Green" title="E24">A12</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:45_1_1_3"/><i><font color="Green" title="E19">A13</font></i>: 
for <font color="Olive" title="b1">m</font> being   <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>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>]
 
<b>from </b><i><a class="ref" href="triang_1.html#S1" target="_self" title="TRIANG_1:sch.1">TRIANG_1:sch 1</a>(<a class="txt" href="triang_1.html#E1:45_1_1_3"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="triang_1.html#E2:45_1_1_3"><i><font color="Green" title="E18">A6</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c4">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<a NAME="E4:45_1_1_3"/><i><font color="Green" title="E20">A14</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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>assume </b><a NAME="E5:45_1_1_3"/>
<font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font>
 ;<br/>

<b>hence </b><a NAME="E6:45_1_1_3"/>
not <font color="Maroon" title="c2">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="triang_1.html#E3:45_1_1_3"><i><font color="Green" title="E19">A13</font></i></a>, <a class="txt" href="triang_1.html#E4:45_1_1_3"><i><font color="Green" title="E20">A14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c3">Sk</font> = <font color="Maroon" title="c2">Sk</font> as  <a href="triang_1.html#V1" title="TRIANG_1:attr.1">lower_non-empty</a> <a href="triang_1.html#NM2" title="TRIANG_1:NM.2">SetSequence</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ex <font color="Olive" title="b2">y</font> being  <a href="triang_1.html#NM5" title="TRIANG_1:NM.5">Face</a> of <font color="Olive" title="b1">n</font> st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp; ( for <font color="Olive" title="b3">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>  st <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>for <font color="Olive" title="b4">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font>  st  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s</font> holds <br/>for <font color="Olive" title="b5">g1</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b5">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><font color="Olive" title="b5">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b4">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">y</font> ) );<br/>
<a NAME="E7:45_1_1"/><i><font color="Green" title="E17">A15</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/>for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="triang_1.html#K6" title="TRIANG_1:func.6">NatEmbSeq</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> holds <br/> ex <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="triang_1.html#K4" title="TRIANG_1:func.4">FuncsSeq</a> <font color="Maroon" title="c3">Sk</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">y</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">i</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_1_4"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c4">i</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/>
<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E3:45_1_1_4"/><i><font color="Green" title="E18">A16</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="triang_1.html#K6" title="TRIANG_1:func.6">NatEmbSeq</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 ;<br/>

<a NAME="E4:45_1_1_4"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> :  <a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Olive" title="b1">f</font> is <a href="goboard1.html#V1" title="GOBOARD1:attr.1">increasing</a> </span>}</span> 
 
<b>by </b><i><a class="ref" href="triang_1.html#D7" target="_self" title="TRIANG_1:def.7">Def7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">f</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span><b> such that </b><br/><a NAME="E6:45_1_1_4"/><i><font color="Green" title="E19">A17</font></i>: 
( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp;  <a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Maroon" title="c7">f</font> is <a href="goboard1.html#V1" title="GOBOARD1:attr.1">increasing</a> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c6">x</font> as   <a href="triang_1.html#NM5" title="TRIANG_1:NM.5">Face</a> of <font color="Maroon" title="c5">n</font> <b>by </b><i><a class="txt" href="triang_1.html#E3:45_1_1_4"><i><font color="Green" title="E18">A16</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">y1</font> = <font color="Maroon" title="c8">y</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<b>consider </b><font color="Maroon" title="c10">y2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E10:45_1_1_4"/><i><font color="Green" title="E20">A18</font></i>: 
( <font color="Maroon" title="c10">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y1</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">y2</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="c5">n</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">y2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="triang_1.html#E6:45_1_1_4"><i><font color="Green" title="E19">A17</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">y2</font> = <font color="Maroon" title="c10">y2</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E12:45_1_1_4"/><i><font color="Green" title="E21">A19</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font>
 
<b>by </b><i><a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">y2</font> = <font color="Maroon" title="c11">y2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <b>by </b><i><a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E14:45_1_1_4"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E15:45_1_1_4"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">y2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">y3</font> = <font color="Maroon" title="c12">y2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E17:45_1_1_4"/><i><font color="Green" title="E22">A20</font></i>: 
<font color="Maroon" title="c12">y2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c15">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:45_1_1_4_1"/><i><font color="Green" title="E23">A21</font></i>: 
( <font color="Maroon" title="c14">a</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> <font color="Maroon" title="c12">y2</font> &amp; <font color="Maroon" title="c15">b</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> <font color="Maroon" title="c12">y2</font> &amp; <font color="Maroon" title="c12">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">b</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c16">a</font> = <font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c17">b</font> = <font color="Maroon" title="c15">b</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_4_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_1_4_1_1"/><i><font color="Green" title="E24">A22</font></i>: 
<font color="Maroon" title="c16">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">b</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_1_4_1_1_1"/>
( <font color="Maroon" title="c16">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">b</font> or <font color="Maroon" title="c17">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">a</font> )
 </span><b>by </b><i><a class="txt" href="triang_1.html#E1:45_1_1_4_1_1"><i><font color="Green" title="E24">A22</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_1_4_1_1_1_1"/>
<font color="Maroon" title="c16">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:45_1_1_4_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="triang_1.html#E6:45_1_1_4"><i><font color="Green" title="E19">A17</font></i></a>, <a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="triang_1.html#E1:45_1_1_4_1"><i><font color="Green" title="E23">A21</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_1_4_1_1_1_2"/>
<font color="Maroon" title="c17">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">a</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:45_1_1_4_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="triang_1.html#E6:45_1_1_4"><i><font color="Green" title="E19">A17</font></i></a>, <a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="triang_1.html#E1:45_1_1_4_1"><i><font color="Green" title="E23">A21</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:45_1_1_4_1"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b</font>
 ;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">y1</font> );<br/>
<a NAME="E18:45_1_1_4"/><i><font color="Green" title="E23">A23</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/> ex <font color="Olive" title="b2">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_1_4_2"/>
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:45_1_1_4_2"/><b>then </b>
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">s'</font></span>)</span> where <font color="Olive" title="b1">s'</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 </span>}</span> 
 
<b>by </b><i><a class="txt" href="triang_1.html#E4:45_1_1"><i><font color="Green" title="E16">A3</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">A</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E4:45_1_1_4_2"/><i><font color="Green" title="E24">A24</font></i>: 
(  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c15">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c15">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c16">u</font> = <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c15">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">y</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>consider </b><font color="Maroon" title="c17">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E7:45_1_1_4_2"/><i><font color="Green" title="E25">A25</font></i>: 
<font color="Maroon" title="c17">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E4:45_1_1_4_2"><i><font color="Green" title="E24">A24</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c16">u</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c17">f</font>
;<br/>

<b>thus </b><a NAME="E8:45_1_1_4_2"/>
( <font color="Maroon" title="c17">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s</font> &amp; <font color="Maroon" title="c16">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">y1</font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E4:45_1_1_4_2"><i><font color="Green" title="E24">A24</font></i></a>, <a class="txt" href="triang_1.html#E7:45_1_1_4_2"><i><font color="Green" title="E25">A25</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c14">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E20:45_1_1_4"/><i><font color="Green" title="E24">A26</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E21:45_1_1_4"/><i><font color="Green" title="E25">A27</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font>]
 <b>from </b><i><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="triang_1.html#E18:45_1_1_4"><i><font color="Green" title="E23">A23</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c15">g'</font> = <font color="Maroon" title="c14">g</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c15">g'</font>
;<br/>

<a NAME="E23:45_1_1_4"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:45_1_1_4_3"/>
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:45_1_1_4_3"/><i><font color="Green" title="E26">A28</font></i>: 
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">g</font> &amp; <font color="Maroon" title="c16">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:45_1_1_4_3"/><i><font color="Green" title="E27">A29</font></i>: 
( <font color="Maroon" title="c18">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font> &amp; <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">y2</font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="triang_1.html#E20:45_1_1_4"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="triang_1.html#E21:45_1_1_4"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="triang_1.html#E3:45_1_1_4_3"><i><font color="Green" title="E26">A28</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">F</font> =  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font> as  <a href="setfam_1.html#NV3" title="SETFAM_1:NV.3">with_non-empty_element</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_4_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_1_4_3_1"/>
( <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_4_3_1_1"><b>suppose </b></a><a NAME="E1:45_1_1_4_3_1_1"/><i><font color="Green" title="E28">A30</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:45_1_1_4_3_1_1"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">y1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="relat_1.html#T44" title="RELAT_1:th.44">RELAT_1:44</a></i>;<br/><a NAME="E3:45_1_1_4_3_1_1"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">y1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><a NAME="E4:45_1_1_4_3_1_1"/><b>then </b>
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="triang_1.html#E3:45_1_1_4_3"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="triang_1.html#E5:45_1_1_4_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/><b>hence </b><a NAME="E5:45_1_1_4_3_1_1"/>
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E3:45_1_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="triang_1.html#E1:45_1_1_4_3_1_1"><i><font color="Green" title="E28">A30</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_4_3_1_2"><b>suppose </b></a><a NAME="E1:45_1_1_4_3_1_2"/><i><font color="Green" title="E28">A31</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:45_1_1_4_3_1_2"/>
<font color="Maroon" title="c18">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">s1</font></span>)</span> where <font color="Olive" title="b1">s1</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 </span>}</span> 
 <b>by </b><i><a class="txt" href="triang_1.html#E4:45_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="triang_1.html#E20:45_1_1_4"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="triang_1.html#E3:45_1_1_4_3"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="triang_1.html#E5:45_1_1_4_3"><i><font color="Green" title="E27">A29</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">s1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E4:45_1_1_4_3_1_2"/><i><font color="Green" title="E29">A32</font></i>: 
(  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c20">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c20">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><a NAME="E5:45_1_1_4_3_1_2"/>
<font color="Maroon" title="c20">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> : the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Olive" title="b1">A</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c21">s1'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E7:45_1_1_4_3_1_2"/><i><font color="Green" title="E30">A33</font></i>: 
( <font color="Maroon" title="c21">s1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">s1</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Maroon" title="c21">s1'</font> )
 ;<br/><a NAME="E8:45_1_1_4_3_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">s1</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E4:45_1_1_4_3_1_2"><i><font color="Green" title="E29">A32</font></i></a>, <a class="txt" href="triang_1.html#E7:45_1_1_4_3_1_2"><i><font color="Green" title="E30">A33</font></i></a>, <a class="ref" href="triang_1.html#D2" target="_self" title="TRIANG_1:def.2">Def2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c22">f</font> = <font color="Maroon" title="c18">f</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c20">s1</font> <b>by </b><i><a class="txt" href="triang_1.html#E4:45_1_1_4_3_1_2"><i><font color="Green" title="E29">A32</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">f</font> = <font color="Maroon" title="c22">f</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c20">s1</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font> <a href="triang_1.html#K1" title="TRIANG_1:func.1">|_2</a> <font color="Maroon" title="c20">s1</font></span>)</span> #) ;<br/><a NAME="E11:45_1_1_4_3_1_2"/><i><font color="Green" title="E31">A34</font></i>: 
<font color="Maroon" title="c23">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="triang_1.html#E4:45_1_1_4_3_1_2"><i><font color="Green" title="E29">A32</font></i></a>, <a class="txt" href="triang_1.html#E7:45_1_1_4_3_1_2"><i><font color="Green" title="E30">A33</font></i></a>, <a class="ref" href="triang_1.html#T8" target="_self" title="TRIANG_1:th.8">Th8</a></i>;<br/><a NAME="E12:45_1_1_4_3_1_2"/><i><font color="Green" title="E32">A35</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">f</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"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="triang_1.html#E4:45_1_1_4_3_1_2"><i><font color="Green" title="E29">A32</font></i></a>, <a class="ref" href="triang_1.html#T10" target="_self" title="TRIANG_1:th.10">Th10</a></i>;<br/><a NAME="E13:45_1_1_4_3_1_2"/><i><font color="Green" title="E33">A36</font></i>: 
<font color="Maroon" title="c23">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">f</font>,<font color="Maroon" title="c20">s1</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T30" title="FINSEQ_2:th.30">FINSEQ_2:30</a></i>;<br/><a NAME="E14:45_1_1_4_3_1_2"/><i><font color="Green" title="E34">A37</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c23">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">s1</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><a NAME="E15:45_1_1_4_3_1_2"/>
<font color="Maroon" title="c23">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E12:45_1_1_4_3_1_2"><i><font color="Green" title="E32">A35</font></i></a>, <a class="txt" href="triang_1.html#E13:45_1_1_4_3_1_2"><i><font color="Green" title="E33">A36</font></i></a>, <a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c24">z1</font> = <font color="Maroon" title="c16">z</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font> #) <b>by </b><i><a class="txt" href="triang_1.html#E3:45_1_1_4_3"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="triang_1.html#E5:45_1_1_4_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="ref" href="finseq_2.html#T36" title="FINSEQ_2:th.36">FINSEQ_2:36</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">f</font> = <font color="Maroon" title="c23">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="txt" href="triang_1.html#E12:45_1_1_4_3_1_2"><i><font color="Green" title="E32">A35</font></i></a>, <a class="txt" href="triang_1.html#E13:45_1_1_4_3_1_2"><i><font color="Green" title="E33">A36</font></i></a>, <a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></i>;<br/><a NAME="E18:45_1_1_4_3_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">f</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c12">y2</font></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 <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c26">r</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">f</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c12">y2</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/><a NAME="E20:45_1_1_4_3_1_2"/><i><font color="Green" title="E35">A38</font></i>: 
<font color="Maroon" title="c5">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> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E1:45_1_1_4_3_1_2"><i><font color="Green" title="E28">A31</font></i></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/><a NAME="E21:45_1_1_4_3_1_2"/><i><font color="Green" title="E36">A39</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">f</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c12">y2</font></span>)</span> <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="c5">n</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="triang_1.html#E12:45_1_1_4_3_1_2"><i><font color="Green" title="E32">A35</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/><a NAME="E22:45_1_1_4_3_1_2"/>
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">f</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c12">y2</font></span>)</span> holds <br/><font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T25" title="FUNCT_1:th.25">FUNCT_1:25</a></i>;<br/><a NAME="E23:45_1_1_4_3_1_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">f</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c12">y2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E24:45_1_1_4_3_1_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">f</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c12">y2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">s1</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E14:45_1_1_4_3_1_2"><i><font color="Green" title="E34">A37</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c27">s'</font> = <font color="Maroon" title="c26">r</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of <font color="Maroon" title="c19">F</font> <b>by </b><i><a class="txt" href="triang_1.html#E20:45_1_1_4_3_1_2"><i><font color="Green" title="E35">A38</font></i></a>, <a class="txt" href="triang_1.html#E21:45_1_1_4_3_1_2"><i><font color="Green" title="E36">A39</font></i></a>, <a class="ref" href="triang_1.html#T7" target="_self" title="TRIANG_1:th.7">Th7</a>, <a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/><a NAME="E26:45_1_1_4_3_1_2"/>
for <font color="Olive" title="b1">n1</font>, <font color="Olive" title="b2">m1</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n1</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> <font color="Maroon" title="c24">z1</font> &amp; <font color="Olive" title="b2">m1</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> <font color="Maroon" title="c24">z1</font> &amp; <font color="Olive" title="b1">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m1</font> holds <br/>( <font color="Maroon" title="c24">z1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">z1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">m1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">z1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">z1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">m1</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_4_3_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c28">n1</font>, <font color="Maroon" title="c29">m1</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:45_1_1_4_3_1_2_1"/><i><font color="Green" title="E37">A40</font></i>: 
( <font color="Maroon" title="c28">n1</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> <font color="Maroon" title="c24">z1</font> &amp; <font color="Maroon" title="c29">m1</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> <font color="Maroon" title="c24">z1</font> &amp; <font color="Maroon" title="c28">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">m1</font> )
 ;<br/>

<a NAME="E2:45_1_1_4_3_1_2_1"/><b>then </b><i><font color="Green" title="E38">A41</font></i>: 
( <font color="Maroon" title="c24">z1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">n1</font></span>)</span> &amp; <font color="Maroon" title="c24">z1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">m1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="triang_1.html#E3:45_1_1_4_3"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="triang_1.html#E5:45_1_1_4_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<a NAME="E3:45_1_1_4_3_1_2_1"/>
<font color="Maroon" title="c12">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">n1</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"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="triang_1.html#E3:45_1_1_4_3"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="triang_1.html#E5:45_1_1_4_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="triang_1.html#E21:45_1_1_4_3_1_2"><i><font color="Green" title="E36">A39</font></i></a>, <a class="txt" href="triang_1.html#E1:45_1_1_4_3_1_2_1"><i><font color="Green" title="E37">A40</font></i></a>, <a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c30">yn</font> = <font color="Maroon" title="c12">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">n1</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="E5:45_1_1_4_3_1_2_1"/>
<font color="Maroon" title="c12">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">m1</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"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="triang_1.html#E3:45_1_1_4_3"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="triang_1.html#E5:45_1_1_4_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="triang_1.html#E21:45_1_1_4_3_1_2"><i><font color="Green" title="E36">A39</font></i></a>, <a class="txt" href="triang_1.html#E1:45_1_1_4_3_1_2_1"><i><font color="Green" title="E37">A40</font></i></a>, <a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c31">ym</font> = <font color="Maroon" title="c12">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">m1</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="E7:45_1_1_4_3_1_2_1"/><i><font color="Green" title="E39">A42</font></i>: 
<font color="Maroon" title="c30">yn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c31">ym</font>
 
<b>by </b><i><a class="txt" href="triang_1.html#E6:45_1_1_4"><i><font color="Green" title="E19">A17</font></i></a>, <a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="triang_1.html#E3:45_1_1_4_3"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="triang_1.html#E5:45_1_1_4_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="triang_1.html#E21:45_1_1_4_3_1_2"><i><font color="Green" title="E36">A39</font></i></a>, <a class="txt" href="triang_1.html#E1:45_1_1_4_3_1_2_1"><i><font color="Green" title="E37">A40</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E8:45_1_1_4_3_1_2_1"/><i><font color="Green" title="E40">A43</font></i>: 
( <font color="Maroon" title="c30">yn</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">y2</font> &amp; <font color="Maroon" title="c31">ym</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">y2</font> )
 
<b>by </b><i><a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="triang_1.html#E3:45_1_1_4_3"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="triang_1.html#E5:45_1_1_4_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="triang_1.html#E21:45_1_1_4_3_1_2"><i><font color="Green" title="E36">A39</font></i></a>, <a class="txt" href="triang_1.html#E1:45_1_1_4_3_1_2_1"><i><font color="Green" title="E37">A40</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c32">fn</font> = <font color="Maroon" title="c25">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">yn</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c20">s1</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font> <a href="triang_1.html#K1" title="TRIANG_1:func.1">|_2</a> <font color="Maroon" title="c20">s1</font></span>)</span> #) <b>by </b><i><a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="triang_1.html#E12:45_1_1_4_3_1_2"><i><font color="Green" title="E32">A35</font></i></a>, <a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c33">fm</font> = <font color="Maroon" title="c25">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">ym</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c20">s1</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font> <a href="triang_1.html#K1" title="TRIANG_1:func.1">|_2</a> <font color="Maroon" title="c20">s1</font></span>)</span> #) <b>by </b><i><a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="triang_1.html#E12:45_1_1_4_3_1_2"><i><font color="Green" title="E32">A35</font></i></a>, <a class="txt" href="triang_1.html#E8:45_1_1_4_3_1_2_1"><i><font color="Green" title="E40">A43</font></i></a>, <a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E11:45_1_1_4_3_1_2_1"/>
<font color="Maroon" title="c24">z1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">fn</font>
 
<b>by </b><i><a class="txt" href="triang_1.html#E3:45_1_1_4_3"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="triang_1.html#E5:45_1_1_4_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="triang_1.html#E1:45_1_1_4_3_1_2_1"><i><font color="Green" title="E37">A40</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c34">zn</font> = <font color="Maroon" title="c24">z1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">n1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c20">s1</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font> <a href="triang_1.html#K1" title="TRIANG_1:func.1">|_2</a> <font color="Maroon" title="c20">s1</font></span>)</span> #) ;<br/>
<a NAME="E13:45_1_1_4_3_1_2_1"/>
<font color="Maroon" title="c24">z1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">fm</font>
 
<b>by </b><i><a class="txt" href="triang_1.html#E3:45_1_1_4_3"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="triang_1.html#E5:45_1_1_4_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="triang_1.html#E1:45_1_1_4_3_1_2_1"><i><font color="Green" title="E37">A40</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c35">zm</font> = <font color="Maroon" title="c24">z1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">m1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c20">s1</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font> <a href="triang_1.html#K1" title="TRIANG_1:func.1">|_2</a> <font color="Maroon" title="c20">s1</font></span>)</span> #) ;<br/>
<a NAME="E15:45_1_1_4_3_1_2_1"/><i><font color="Green" title="E41">A44</font></i>: 
( <font color="Maroon" title="c32">fn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c30">yn</font> &amp; <font color="Maroon" title="c33">fm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">ym</font> )
 
<b>by </b><i><a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="triang_1.html#E12:45_1_1_4_3_1_2"><i><font color="Green" title="E32">A35</font></i></a>, <a class="txt" href="triang_1.html#E8:45_1_1_4_3_1_2_1"><i><font color="Green" title="E40">A43</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E16:45_1_1_4_3_1_2_1"/><i><font color="Green" title="E42">A45</font></i>: 
( <font color="Maroon" title="c34">zn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">z1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c28">n1</font> &amp; <font color="Maroon" title="c35">zm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">z1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c29">m1</font> )
 
<b>by </b><i><a class="txt" href="triang_1.html#E1:45_1_1_4_3_1_2_1"><i><font color="Green" title="E37">A40</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c36">R</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>;<br/>
<i><font color="Green" title="E43">A46</font></i>: <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c30">yn</font> = 
<font color="Maroon" title="c25">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">yn</font>
<b>by </b><i><a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="triang_1.html#E12:45_1_1_4_3_1_2"><i><font color="Green" title="E32">A35</font></i></a>, <a class="txt" href="triang_1.html#E8:45_1_1_4_3_1_2_1"><i><font color="Green" title="E40">A43</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c20">s1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c30">yn</font>
<b>by </b><i><a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="triang_1.html#E4:45_1_1_4_3_1_2"><i><font color="Green" title="E29">A32</font></i></a>, <a class="txt" href="triang_1.html#E12:45_1_1_4_3_1_2"><i><font color="Green" title="E32">A35</font></i></a>, <a class="txt" href="triang_1.html#E8:45_1_1_4_3_1_2_1"><i><font color="Green" title="E40">A43</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/>
<font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">ym</font> = 
<font color="Maroon" title="c25">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">ym</font>
<b>by </b><i><a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="triang_1.html#E12:45_1_1_4_3_1_2"><i><font color="Green" title="E32">A35</font></i></a>, <a class="txt" href="triang_1.html#E8:45_1_1_4_3_1_2_1"><i><font color="Green" title="E40">A43</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c20">s1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">ym</font>
<b>by </b><i><a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="triang_1.html#E4:45_1_1_4_3_1_2"><i><font color="Green" title="E29">A32</font></i></a>, <a class="txt" href="triang_1.html#E12:45_1_1_4_3_1_2"><i><font color="Green" title="E32">A35</font></i></a>, <a class="txt" href="triang_1.html#E8:45_1_1_4_3_1_2_1"><i><font color="Green" title="E40">A43</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E19:45_1_1_4_3_1_2_1"/>
( <font color="Maroon" title="c24">z1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c28">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">z1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c29">m1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">z1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c28">n1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">z1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c29">m1</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="triang_1.html#E4:45_1_1_4_3_1_2"><i><font color="Green" title="E29">A32</font></i></a>, <a class="txt" href="triang_1.html#E7:45_1_1_4_3_1_2"><i><font color="Green" title="E30">A33</font></i></a>, <a class="txt" href="triang_1.html#E12:45_1_1_4_3_1_2"><i><font color="Green" title="E32">A35</font></i></a>, <a class="txt" href="triang_1.html#E2:45_1_1_4_3_1_2_1"><i><font color="Green" title="E38">A41</font></i></a>, <a class="txt" href="triang_1.html#E7:45_1_1_4_3_1_2_1"><i><font color="Green" title="E39">A42</font></i></a>, <a class="txt" href="triang_1.html#E8:45_1_1_4_3_1_2_1"><i><font color="Green" title="E40">A43</font></i></a>, <a class="txt" href="triang_1.html#E15:45_1_1_4_3_1_2_1"><i><font color="Green" title="E41">A44</font></i></a>, <a class="txt" href="triang_1.html#E16:45_1_1_4_3_1_2_1"><i><font color="Green" title="E42">A45</font></i></a>, <a class="txt" href="triang_1.html#E17:45_1_1_4_3_1_2_1"><i><font color="Green" title="E43">A46</font></i></a>, <a class="ref" href="triang_1.html#D2" target="_self" title="TRIANG_1:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E27:45_1_1_4_3_1_2"/><b>then </b><i><font color="Green" title="E37">A47</font></i>: 
<font color="Maroon" title="c24">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c27">s'</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E3:45_1_1_4_3"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="triang_1.html#E5:45_1_1_4_3"><i><font color="Green" title="E27">A29</font></i></a>, <a class="ref" href="triang_1.html#T4" target="_self" title="TRIANG_1:th.4">Th4</a></i>;<br/><a NAME="E28:45_1_1_4_3_1_2"/><i><font color="Green" title="E38">A48</font></i>: 
<font color="Maroon" title="c25">f</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c12">y2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="triang_1.html#E17:45_1_1_4"><i><font color="Green" title="E22">A20</font></i></a>, <a class="txt" href="triang_1.html#E11:45_1_1_4_3_1_2"><i><font color="Green" title="E31">A34</font></i></a>, <a class="ref" href="funct_1.html#T46" title="FUNCT_1:th.46">FUNCT_1:46</a></i>;<br/><a NAME="E29:45_1_1_4_3_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">f</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c12">y2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E10:45_1_1_4"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="triang_1.html#E12:45_1_1_4"><i><font color="Green" title="E21">A19</font></i></a>, <a class="txt" href="triang_1.html#E12:45_1_1_4_3_1_2"><i><font color="Green" title="E32">A35</font></i></a>, <a class="ref" href="finseq_2.html#T33" title="FINSEQ_2:th.33">FINSEQ_2:33</a></i>;<br/><a NAME="E30:45_1_1_4_3_1_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c27">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E28:45_1_1_4_3_1_2"><i><font color="Green" title="E38">A48</font></i></a>, <a class="ref" href="finseq_4.html#T77" title="FINSEQ_4:th.77">FINSEQ_4:77</a></i>;<br/><a NAME="E31:45_1_1_4_3_1_2"/><b>then </b>
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">s</font></span>)</span> where <font color="Olive" title="b1">s</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="triang_1.html#E27:45_1_1_4_3_1_2"><i><font color="Green" title="E37">A47</font></i></a></i>;<br/><b>hence </b><a NAME="E32:45_1_1_4_3_1_2"/>
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E4:45_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="triang_1.html#E1:45_1_1_4_3_1_2"><i><font color="Green" title="E28">A31</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E24:45_1_1_4"/><b>then </b>
<font color="Maroon" title="c15">g'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="triang_1.html#E20:45_1_1_4"><i><font color="Green" title="E24">A26</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E25:45_1_1_4"/>
<font color="Maroon" title="c15">g'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="triang_1.html#K4" title="TRIANG_1:func.4">FuncsSeq</a> <font color="Maroon" title="c3">Sk</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="ref" href="triang_1.html#D5" target="_self" title="TRIANG_1:def.5">Def5</a></i>;<br/>

<b>take </b>
<font color="Maroon" title="c5">n</font>
;<br/>

<b>reconsider </b><font color="Maroon" title="c16">y</font> = <font color="Maroon" title="c6">x</font> as   <a href="triang_1.html#NM5" title="TRIANG_1:NM.5">Face</a> of <font color="Maroon" title="c5">n</font> <b>by </b><i><a class="txt" href="triang_1.html#E3:45_1_1_4"><i><font color="Green" title="E18">A16</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c16">y</font>
;<br/>

<b>thus </b><a NAME="E27:45_1_1_4"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y</font> &amp; <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c17">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>

<b>assume </b><a NAME="E28:45_1_1_4"/><i><font color="Green" title="E26">A49</font></i>: 
<font color="Maroon" title="c17">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c18">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font>;<br/>

<b>assume </b><a NAME="E29:45_1_1_4"/><i><font color="Green" title="E27">A50</font></i>: 
 <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c18">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">s</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c19">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E31:45_1_1_4"/><i><font color="Green" title="E28">A51</font></i>: 
( <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">s</font> &amp; <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">y1</font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E21:45_1_1_4"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="triang_1.html#E28:45_1_1_4"><i><font color="Green" title="E26">A49</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c20">g1</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>

<b>assume </b><a NAME="E32:45_1_1_4"/>
<font color="Maroon" title="c20">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g'</font>
 ;<br/>

<b>hence </b><a NAME="E33:45_1_1_4"/>
<font color="Maroon" title="c20">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c18">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c16">y</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E29:45_1_1_4"><i><font color="Green" title="E27">A50</font></i></a>, <a class="txt" href="triang_1.html#E31:45_1_1_4"><i><font color="Green" title="E28">A51</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">F</font> being    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of  <a href="triang_1.html#K6" title="TRIANG_1:func.6">NatEmbSeq</a> , <a href="triang_1.html#K4" title="TRIANG_1:func.4">FuncsSeq</a> <font color="Maroon" title="c3">Sk</font><b> such that </b><br/><a NAME="E9:45_1_1"/><i><font color="Green" title="E18">A52</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ex <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="triang_1.html#K6" title="TRIANG_1:func.6">NatEmbSeq</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><a href="triang_1.html#K4" title="TRIANG_1:func.4">FuncsSeq</a> <font color="Maroon" title="c3">Sk</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> st <br/>( <font color="Olive" title="b2">f</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> <font color="Olive" title="b1">i</font> &amp; ( for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="triang_1.html#K6" title="TRIANG_1:func.6">NatEmbSeq</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">i</font>] ) )
 <b>from </b><i><a class="ref" href="mssubfam.html#S1" title="MSSUBFAM:sch.1">MSSUBFAM:sch 1</a>(<a class="txt" href="triang_1.html#E7:45_1_1"><i><font color="Green" title="E17">A15</font></i></a>);<br/></i>
<b>take </b>
 <a href="triang_1.html#G1" title="TRIANG_1:aggr.1">TriangStr</a>(# <font color="Maroon" title="c3">Sk</font>,<font color="Maroon" title="c4">F</font> #)
;<br/>

<b>thus </b><a NAME="E10:45_1_1"/>
the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <a href="triang_1.html#G1" title="TRIANG_1:aggr.1">TriangStr</a>(# <font color="Maroon" title="c3">Sk</font>,<font color="Maroon" title="c4">F</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="triang_1.html#E3:45_1_1"><i><font color="Green" title="E15">A2</font></i></a></i>;<br/>

<b>thus </b><a NAME="E11:45_1_1"/>
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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 holds <br/>the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <a href="triang_1.html#G1" title="TRIANG_1:aggr.1">TriangStr</a>(# <font color="Maroon" title="c3">Sk</font>,<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b2">s</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="triang_1.html#E4:45_1_1"><i><font color="Green" title="E16">A3</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c5">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<a NAME="E12:45_1_1"/>
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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>then consider </b><font color="Maroon" title="c6">f1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="triang_1.html#K6" title="TRIANG_1:func.6">NatEmbSeq</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>,<span class="p1">(<span class="default"><a href="triang_1.html#K4" title="TRIANG_1:func.4">FuncsSeq</a> <font color="Maroon" title="c3">Sk</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font><b> such that </b><br/><a NAME="E14:45_1_1"/><i><font color="Green" title="E19">A53</font></i>: 
<font color="Maroon" title="c6">f1</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> <font color="Maroon" title="c5">n</font>
 <b>and </b><br/><a NAME="E15:45_1_1"/><i><font color="Green" title="E20">A54</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="triang_1.html#K6" title="TRIANG_1:func.6">NatEmbSeq</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> holds <br/> ex <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ex <font color="Olive" title="b3">y</font> being  <a href="triang_1.html#NM5" title="TRIANG_1:NM.5">Face</a> of <font color="Olive" title="b2">m</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y</font> &amp; <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> &amp; ( for <font color="Olive" title="b4">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>  st <font color="Olive" title="b4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>for <font color="Olive" title="b5">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font>  st  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">s</font> holds <br/>for <font color="Olive" title="b6">g1</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> holds <br/><font color="Olive" title="b6">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b5">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b3">y</font> ) )
 <b>by </b><i><a class="txt" href="triang_1.html#E9:45_1_1"><i><font color="Green" title="E18">A52</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">x</font> be   <a href="triang_1.html#NM5" title="TRIANG_1:NM.5">Face</a> of <font color="Maroon" title="c5">n</font>;<br/><b>let </b><font color="Maroon" title="c8">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <a href="triang_1.html#G1" title="TRIANG_1:aggr.1">TriangStr</a>(# <font color="Maroon" title="c3">Sk</font>,<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="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>;<br/>



<b>assume </b><a NAME="E16:45_1_1"/><i><font color="Green" title="E21">A55</font></i>: 
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <a href="triang_1.html#G1" title="TRIANG_1:aggr.1">TriangStr</a>(# <font color="Maroon" title="c3">Sk</font>,<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="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c9">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font>;<br/>

<b>assume </b><a NAME="E17:45_1_1"/><i><font color="Green" title="E22">A56</font></i>: 
 <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c10">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>, <font color="Maroon" title="c11">y</font> being   <a href="triang_1.html#NM5" title="TRIANG_1:NM.5">Face</a> of <font color="Maroon" title="c10">m</font><b> such that </b><br/><a NAME="E19:45_1_1"/><i><font color="Green" title="E23">A57</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> &amp; <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font> &amp; ( for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>for <font color="Olive" title="b2">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font>  st  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> holds <br/>for <font color="Olive" title="b3">g1</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b3">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> holds <br/><font color="Olive" title="b3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b2">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">y</font> ) )
 <b>by </b><i><a class="txt" href="triang_1.html#E15:45_1_1"><i><font color="Green" title="E20">A54</font></i></a></i>;<br/>
<a NAME="E20:45_1_1"/>
<font color="Maroon" title="c6">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="triang_1.html#K4" title="TRIANG_1:func.4">FuncsSeq</a> <font color="Maroon" title="c3">Sk</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
 
;<br/>
<a NAME="E21:45_1_1"/><b>then </b>
<font color="Maroon" title="c6">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="triang_1.html#D5" target="_self" title="TRIANG_1:def.5">Def5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">G</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E23:45_1_1"/><i><font color="Green" title="E24">A58</font></i>: 
<font color="Maroon" title="c6">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">G</font>
 <b>and </b><br/><a NAME="E24:45_1_1"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Sk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E25:45_1_1"/>
 <a href="triang_1.html#K7" title="TRIANG_1:func.7">face</a> <font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s</font>
 
<b>by </b><i><a class="txt" href="triang_1.html#E14:45_1_1"><i><font color="Green" title="E19">A53</font></i></a>, <a class="txt" href="triang_1.html#E16:45_1_1"><i><font color="Green" title="E21">A55</font></i></a>, <a class="txt" href="triang_1.html#E23:45_1_1"><i><font color="Green" title="E24">A58</font></i></a>, <a class="ref" href="triang_1.html#D10" target="_self" title="TRIANG_1:def.10">Def10</a></i>;<br/>
<b>hence </b><a NAME="E26:45_1_1"/>
 <a href="triang_1.html#K7" title="TRIANG_1:func.7">face</a> <font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E16:45_1_1"><i><font color="Green" title="E21">A55</font></i></a>, <a class="txt" href="triang_1.html#E17:45_1_1"><i><font color="Green" title="E22">A56</font></i></a>, <a class="txt" href="triang_1.html#E19:45_1_1"><i><font color="Green" title="E23">A57</font></i></a>, <a class="txt" href="triang_1.html#E23:45_1_1"><i><font color="Green" title="E24">A58</font></i></a></i>;<br/>


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