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

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">a<sub>1</sub></font>,0,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="triang_1.html#M1" target="_self">Element</a> of  <a href="triang_1.html#K3" target="_self">symplexes</a> <font color="Maroon">C</font> :  <a href="card_1.html#K1">Card</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">Sk</font> being   <a href="triang_1.html#NM2">SetSequence</a><b> such that </b><br/><a NAME="E2:45_1_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5">PBOOLE:sch 5</a>();<br/></i>
<i><font color="Green">E35</font></i>: <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> 0 = 
 <a href="cqc_lang.html#K1">IFEQ</a> 0,0,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="triang_1.html#M1" target="_self">Element</a> of  <a href="triang_1.html#K3" target="_self">symplexes</a> <font color="Maroon">C</font> :  <a href="card_1.html#K1">Card</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> 0 </span>}</span> 
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/>
<div><i><font color="Green">E37</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">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:45_1_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:45_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
 <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">n</font>,0,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="triang_1.html#M1" target="_self">Element</a> of  <a href="triang_1.html#K3" target="_self">symplexes</a> <font color="Maroon">C</font> :  <a href="card_1.html#K1">Card</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> </span>}</span> 
<b>by </b><i/>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="triang_1.html#M1" target="_self">Element</a> of  <a href="triang_1.html#K3" target="_self">symplexes</a> <font color="Maroon">C</font> :  <a href="card_1.html#K1">Card</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> </span>}</span> 
<b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E5:45_1_1"/>
<font color="Maroon">Sk</font> is <a href="triang_1.html#V1" target="_self">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">n</font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="triang_1.html#D4" target="_self">TRIANG_1:def 4</a><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>not <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a>;<br/>
<b>assume </b><a NAME="E1:45_1_1_3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<a NAME="E2:45_1_1_3"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">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">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_1_3_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; not <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:45_1_1_3_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:45_1_1_3_1"/>
<font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="triang_1.html#M1" target="_self">Element</a> of  <a href="triang_1.html#K3" target="_self">symplexes</a> <font color="Maroon">C</font> :  <a href="card_1.html#K1">Card</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">s</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="triang_1.html#M1" target="_self">Element</a> of  <a href="triang_1.html#K3" target="_self">symplexes</a> <font color="Maroon">C</font><b> such that </b><br/><a NAME="E6:45_1_1_3_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Maroon">s</font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">s</font>;<br/>
<a NAME="E8:45_1_1_3_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">s</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<b>reconsider </b><font color="Maroon">sx</font> = <font color="Maroon">s</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E10:45_1_1_3_1"/>
<font color="Maroon">sx</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>;<br/>
<a NAME="E11:45_1_1_3_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">sx</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E12:45_1_1_3_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sx</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E13:45_1_1_3_1"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">sx</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="card_2.html#T54">CARD_2:54</a></i>;<br/>
<a NAME="E14:45_1_1_3_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">sx</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
;<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">m</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&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">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:45_1_1_3_1_1_1"/>
not <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><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">E58</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">sx</font> = <font color="Maroon">sx</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="triang_1.html#M1" target="_self">Element</a> of  <a href="triang_1.html#K3" target="_self">symplexes</a> <font color="Maroon">C</font> <b>by </b><i>, , , <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/><a NAME="E3:45_1_1_3_1_1_2"/>
 <a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Maroon">sx</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Olive">s1</font></span>)</span> where <font color="Olive">s1</font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="triang_1.html#M1" target="_self">Element</a> of  <a href="triang_1.html#K3" target="_self">symplexes</a> <font color="Maroon">C</font> :  <a href="card_1.html#K1">Card</a> <font color="Olive">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:45_1_1_3_1_1_2"/>
not <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, </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">E71</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>]
 
<b>from </b><i><a class="ref" href="triang_1.html#S1" target="_self">TRIANG_1:sch 1</a>(, );<br/></i>
<b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E4:45_1_1_3"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E5:45_1_1_3"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/>

<b>hence </b><a NAME="E6:45_1_1_3"/>
not <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Sk</font> = <font color="Maroon">Sk</font> as  <a href="triang_1.html#V1" target="_self">lower_non-empty</a> <a href="triang_1.html#NM2">SetSequence</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> ex <font color="Olive">y</font> being  <a href="triang_1.html#NM5">Face</a> of <font color="Olive">n</font> st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp; ( for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>  st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> holds <br/>for <font color="Olive">A</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="triang_1.html#M1" target="_self">Element</a> of  <a href="triang_1.html#K3" target="_self">symplexes</a> <font color="Maroon">C</font>  st  <a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> holds <br/>for <font color="Olive">g1</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Olive">A</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">y</font> ) );<br/>
<a NAME="E7:45_1_1"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <a href="triang_1.html#K6" target="_self">NatEmbSeq</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="triang_1.html#K4" target="_self">FuncsSeq</a> <font color="Maroon">Sk</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>,<font color="Olive">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">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_1_4"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E3:45_1_1_4"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="triang_1.html#K6" target="_self">NatEmbSeq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E4:45_1_1_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> :  <a href="triang_1.html#K5" target="_self">@</a> <font color="Olive">f</font> is <a href="goboard1.html#V1">increasing</a> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">f</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span><b> such that </b><br/><a NAME="E6:45_1_1_4"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="triang_1.html#K5" target="_self">@</a> <font color="Maroon">f</font> is <a href="goboard1.html#V1">increasing</a> )
 ;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="triang_1.html#NM5">Face</a> of <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<b>consider </b><font color="Maroon">y2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E10:45_1_1_4"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">y2</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">y2</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i>, <a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E12:45_1_1_4"/><i><font color="Green">E77</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">y2</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E14:45_1_1_4"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E15:45_1_1_4"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">y2</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y3</font> = <font color="Maroon">y2</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E17:45_1_1_4"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">y3</font> is <a href="goboard1.html#V1">increasing</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E18:45_1_1_4"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">y2</font> is <a href="funct_1.html#V2">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">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:45_1_1_4_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">y2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">E83</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">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">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> or <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">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">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">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="ref" href="goboard1.html#D1">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">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">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="ref" href="goboard1.html#D1">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">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">y1</font> );<br/>
<a NAME="E19:45_1_1_4"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/> ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">s</font>,<font color="Olive">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">s</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_1_4_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:45_1_1_4_2"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Olive">s'</font></span>)</span> where <font color="Olive">s'</font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="triang_1.html#M1" target="_self">Element</a> of  <a href="triang_1.html#K3" target="_self">symplexes</a> <font color="Maroon">C</font> :  <a href="card_1.html#K1">Card</a> <font color="Olive">s'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">A</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="triang_1.html#M1" target="_self">Element</a> of  <a href="triang_1.html#K3" target="_self">symplexes</a> <font color="Maroon">C</font><b> such that </b><br/><a NAME="E4:45_1_1_4_2"/><i><font color="Green">E88</font></i>: 
(  <a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <span class="p1">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">y</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:45_1_1_4_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">u</font>
;<br/>

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

<b>thus </b><a NAME="E8:45_1_1_4_2"/>
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">y1</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E21:45_1_1_4"/><i><font color="Green">E90</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E22:45_1_1_4"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">s</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">g'</font>
;<br/>

<a NAME="E24:45_1_1_4"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">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">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:45_1_1_4_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:45_1_1_4_3"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:45_1_1_4_3"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">y2</font> )
 <b>by </b><i>, , , </i>;<br/>
<b>reconsider </b><font color="Maroon">F</font> =  <a href="triang_1.html#K3" target="_self">symplexes</a> <font color="Maroon">C</font> as  <a href="setfam_1.html#V2">with_non-empty_element</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&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">E96</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</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">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">y1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="relat_1.html#T44">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">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">y1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E4:45_1_1_4_3_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , , <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/><b>hence </b><a NAME="E5:45_1_1_4_3_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">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">E97</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:45_1_1_4_3_1_2"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Olive">s1</font></span>)</span> where <font color="Olive">s1</font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="triang_1.html#M1" target="_self">Element</a> of  <a href="triang_1.html#K3" target="_self">symplexes</a> <font color="Maroon">C</font> :  <a href="card_1.html#K1">Card</a> <font color="Olive">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 <b>by </b><i>, , , </i>;<br/><b>then consider </b><font color="Maroon">s1</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="triang_1.html#M1" target="_self">Element</a> of  <a href="triang_1.html#K3" target="_self">symplexes</a> <font color="Maroon">C</font><b> such that </b><br/><a NAME="E4:45_1_1_4_3_1_2"/><i><font color="Green">E98</font></i>: 
(  <a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><a NAME="E5:45_1_1_4_3_1_2"/>
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> : the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font> <a href="orders_1.html#R3">linearly_orders</a> <font color="Olive">A</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">s1'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E7:45_1_1_4_3_1_2"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">s1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font> <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">s1'</font> )
 ;<br/><a NAME="E8:45_1_1_4_3_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font>
 <b>by </b><i>, , </i>;<br/><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">s1</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM7">FinSequence</a> of <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">s1</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font> <a href="triang_1.html#K1" target="_self">|_2</a> <font color="Maroon">s1</font></span>)</span> #) ;<br/><a NAME="E11:45_1_1_4_3_1_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, , </i>;<br/><a NAME="E12:45_1_1_4_3_1_2"/><i><font color="Green">E103</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E13:45_1_1_4_3_1_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">f</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>,<font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T30">FINSEQ_2:30</a></i>;<br/><a NAME="E14:45_1_1_4_3_1_2"/><i><font color="Green">E107</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E15:45_1_1_4_3_1_2"/>
<font color="Maroon">f</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>
 <b>by </b><i>, <a class="txt" href="triang_1.html#E2:45_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/><b>then reconsider </b><font color="Maroon">z1</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM7">FinSequence</a> of <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font> #) <b>by </b><i>, , <a class="ref" href="finseq_2.html#T36">FINSEQ_2:36</a></i>;<br/><b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> <b>by </b><i>, <a class="txt" href="triang_1.html#E2:45_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/><a NAME="E18:45_1_1_4_3_1_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">y2</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon">r</font> =  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">y2</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> <b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/><a NAME="E20:45_1_1_4_3_1_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/><a NAME="E21:45_1_1_4_3_1_2"/><i><font color="Green">E109</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">y2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><a NAME="E22:45_1_1_4_3_1_2"/>
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">y2</font></span>)</span> holds <br/><font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T25">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">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">y2</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">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">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">y2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E3:45_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">s'</font> = <font color="Maroon">r</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="triang_1.html#M1" target="_self">Element</a> of <font color="Maroon">F</font> <b>by </b><i>, <a class="txt" href="triang_1.html#E4:45_1_1"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/><a NAME="E26:45_1_1_4_3_1_2"/>
for <font color="Olive">n1</font>, <font color="Olive">m1</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z1</font> &amp; <font color="Olive">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z1</font> &amp; <font color="Olive">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m1</font> holds <br/>( <font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m1</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m1</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">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">n1</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">m1</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:45_1_1_4_3_1_2_1"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z1</font> &amp; <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z1</font> &amp; <font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m1</font> )
 ;<br/>

<a NAME="E2:45_1_1_4_3_1_2_1"/><b>then </b><i><font color="Green">E111</font></i>: 
( <font color="Maroon">z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">y2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font></span>)</span> &amp; <font color="Maroon">z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">y2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E3:45_1_1_4_3_1_2_1"/>
<font color="Maroon">y2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , , <a class="txt" href="triang_1.html#E4:45_1_1"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">yn</font> = <font color="Maroon">y2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E5:45_1_1_4_3_1_2_1"/>
<font color="Maroon">y2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , , <a class="txt" href="triang_1.html#E4:45_1_1"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">ym</font> = <font color="Maroon">y2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E7:45_1_1_4_3_1_2_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">yn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">ym</font>
 
<b>by </b><i>, , , , <a class="txt" href="triang_1.html#E4:45_1_1"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E8:45_1_1_4_3_1_2_1"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">yn</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">ym</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">y2</font> )
 
<b>by </b><i>, , , <a class="txt" href="triang_1.html#E4:45_1_1"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fn</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">yn</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">s1</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font> <a href="triang_1.html#K1" target="_self">|_2</a> <font color="Maroon">s1</font></span>)</span> #) <b>by </b><i>, , , <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fm</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ym</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">s1</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font> <a href="triang_1.html#K1" target="_self">|_2</a> <font color="Maroon">s1</font></span>)</span> #) <b>by </b><i>, , , <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E11:45_1_1_4_3_1_2_1"/>
<font color="Maroon">z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fn</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">zn</font> = <font color="Maroon">z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">s1</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font> <a href="triang_1.html#K1" target="_self">|_2</a> <font color="Maroon">s1</font></span>)</span> #) ;<br/>
<a NAME="E13:45_1_1_4_3_1_2_1"/>
<font color="Maroon">z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fm</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">zm</font> = <font color="Maroon">z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">s1</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font> <a href="triang_1.html#K1" target="_self">|_2</a> <font color="Maroon">s1</font></span>)</span> #) ;<br/>
<a NAME="E15:45_1_1_4_3_1_2_1"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">fn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">yn</font> &amp; <font color="Maroon">fm</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ym</font> )
 
<b>by </b><i>, , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E16:45_1_1_4_3_1_2_1"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">zn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">zm</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m1</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">R</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>;<br/>
<i><font color="Green">E117</font></i>: <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">yn</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">yn</font>
<b>by </b><i>, , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Maroon">s1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">yn</font>
<b>by </b><i>, , , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ym</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ym</font>
<b>by </b><i>, , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Maroon">s1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ym</font>
<b>by </b><i>, , , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E19:45_1_1_4_3_1_2_1"/>
( <font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m1</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m1</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font> )
 <b>by </b><i>, , , , , , , , , , </i>;<br/>


</div><b>end;</b></div><a NAME="E27:45_1_1_4_3_1_2"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a>  <a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Maroon">s'</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E28:45_1_1_4_3_1_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">f</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">y2</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T46">FUNCT_1:46</a></i>;<br/><a NAME="E29:45_1_1_4_3_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">y2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_2.html#T33">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">Card</a> <font color="Maroon">s'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T77">FINSEQ_4:77</a></i>;<br/><a NAME="E31:45_1_1_4_3_1_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="triang_1.html#M1" target="_self">Element</a> of  <a href="triang_1.html#K3" target="_self">symplexes</a> <font color="Maroon">C</font> :  <a href="card_1.html#K1">Card</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E32:45_1_1_4_3_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E25:45_1_1_4"/><b>then </b>
<font color="Maroon">g'</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E26:45_1_1_4"/>
<font color="Maroon">g'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="triang_1.html#K4" target="_self">FuncsSeq</a> <font color="Maroon">Sk</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/>

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

<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="triang_1.html#NM5">Face</a> of <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E28:45_1_1_4"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 ;<br/>

<b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>

<b>assume </b><a NAME="E29:45_1_1_4"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="triang_1.html#M1" target="_self">Element</a> of  <a href="triang_1.html#K3" target="_self">symplexes</a> <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E30:45_1_1_4"/><i><font color="Green">E121</font></i>: 
 <a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/>

<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E32:45_1_1_4"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">y1</font> )
 <b>by </b><i>, </i>;<br/>
<b>let </b><font color="Maroon">g1</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E33:45_1_1_4"/>
<font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font>
 ;<br/>

<b>hence </b><a NAME="E34:45_1_1_4"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being    <a href="pboole.html#M3">ManySortedFunction</a> of  <a href="triang_1.html#K6" target="_self">NatEmbSeq</a> , <a href="triang_1.html#K4" target="_self">FuncsSeq</a> <font color="Maroon">Sk</font><b> such that </b><br/><a NAME="E9:45_1_1"/><i><font color="Green">E123</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/> ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <a href="triang_1.html#K6" target="_self">NatEmbSeq</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>,<span class="p1">(<span class="default"><a href="triang_1.html#K4" target="_self">FuncsSeq</a> <font color="Maroon">Sk</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <a href="triang_1.html#K6" target="_self">NatEmbSeq</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">x</font>,<font color="Olive">i</font>] ) )
 <b>from </b><i><a class="ref" href="mssubfam.html#S1">MSSUBFAM:sch 1</a>();<br/></i>
<b>take </b>
 <a href="triang_1.html#G1" target="_self">TriangStr</a>(# <font color="Maroon">Sk</font>,<font color="Maroon">F</font> #)
;<br/>

<b>thus </b><a NAME="E10:45_1_1"/>
the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <a href="triang_1.html#G1" target="_self">TriangStr</a>(# <font color="Maroon">Sk</font>,<font color="Maroon">F</font> #) <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E11:45_1_1"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/>the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <a href="triang_1.html#G1" target="_self">TriangStr</a>(# <font color="Maroon">Sk</font>,<font color="Maroon">F</font> #) <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="triang_1.html#M1" target="_self">Element</a> of  <a href="triang_1.html#K3" target="_self">symplexes</a> <font color="Maroon">C</font> :  <a href="card_1.html#K1">Card</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> </span>}</span> 
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E12:45_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">f1</font> being   <a href="funct_2.html#NM1">Function</a> of <a href="triang_1.html#K6" target="_self">NatEmbSeq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="triang_1.html#K4" target="_self">FuncsSeq</a> <font color="Maroon">Sk</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E14:45_1_1"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E15:45_1_1"/><i><font color="Green">E125</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <a href="triang_1.html#K6" target="_self">NatEmbSeq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> holds <br/> ex <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a> ex <font color="Olive">y</font> being  <a href="triang_1.html#NM5">Face</a> of <font color="Olive">m</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> &amp; ( for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>  st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> holds <br/>for <font color="Olive">A</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="triang_1.html#M1" target="_self">Element</a> of  <a href="triang_1.html#K3" target="_self">symplexes</a> <font color="Maroon">C</font>  st  <a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> holds <br/>for <font color="Olive">g1</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> holds <br/><font color="Olive">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Olive">A</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">y</font> ) )
 <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="triang_1.html#NM5">Face</a> of <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <a href="triang_1.html#G1" target="_self">TriangStr</a>(# <font color="Maroon">Sk</font>,<font color="Maroon">F</font> #) <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>;<br/>



<b>assume </b><a NAME="E16:45_1_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <a href="triang_1.html#G1" target="_self">TriangStr</a>(# <font color="Maroon">Sk</font>,<font color="Maroon">F</font> #) <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="triang_1.html#M1" target="_self">Element</a> of  <a href="triang_1.html#K3" target="_self">symplexes</a> <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E17:45_1_1"/><i><font color="Green">E127</font></i>: 
 <a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/>

<b>consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">y</font> being   <a href="triang_1.html#NM5">Face</a> of <font color="Maroon">m</font><b> such that </b><br/><a NAME="E19:45_1_1"/><i><font color="Green">E128</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; ( for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>  st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> holds <br/>for <font color="Olive">A</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="triang_1.html#M1" target="_self">Element</a> of  <a href="triang_1.html#K3" target="_self">symplexes</a> <font color="Maroon">C</font>  st  <a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> holds <br/>for <font color="Olive">g1</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> holds <br/><font color="Olive">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Olive">A</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">y</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E20:45_1_1"/>
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="triang_1.html#K4" target="_self">FuncsSeq</a> <font color="Maroon">Sk</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 
;<br/>
<a NAME="E21:45_1_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">G</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E23:45_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E24:45_1_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Sk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E25:45_1_1"/>
 <a href="triang_1.html#K7" target="_self">face</a> <font color="Maroon">s</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E26:45_1_1"/>
 <a href="triang_1.html#K7" target="_self">face</a> <font color="Maroon">s</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" target="_self">SgmX</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , </i>;<br/>


</div>
