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

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="triang_1.html#V2" target="_self">strict</a> <a href="triang_1.html#V3" target="_self">lower_non-empty</a>  <a href="triang_1.html#L1" target="_self">TriangStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:45_1_2"/><i><font color="Green">E14</font></i>: 
the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>2</sub></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>and </b><br/><a NAME="E2:45_1_2"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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 <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> where B 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<sub>1</sub></font> :  <a href="card_1.html#K1">Card</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>and </b><br/><a NAME="E3:45_1_2"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="triang_1.html#NM5">Face</a> of <font color="Olive">b<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/>for <font color="Olive">b<sub>4</sub></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<sub>1</sub></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<sub>1</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> holds <br/> <a href="triang_1.html#K7" target="_self">face</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Olive">b<sub>4</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">b<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:45_1_2"/><i><font color="Green">E17</font></i>: 
the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>3</sub></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>and </b><br/><a NAME="E5:45_1_2"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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 <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> where B 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<sub>1</sub></font> :  <a href="card_1.html#K1">Card</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>and </b><br/><a NAME="E6:45_1_2"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="triang_1.html#NM5">Face</a> of <font color="Olive">b<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/>for <font color="Olive">b<sub>4</sub></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<sub>1</sub></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<sub>1</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> holds <br/> <a href="triang_1.html#K7" target="_self">face</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Olive">b<sub>4</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>

<a NAME="E7:45_1_2"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/>the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_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_2_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>5</sub></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_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:45_1_2_1_1_1_1"/>
the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="triang_1.html#E1:45_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="triang_1.html#E4:45_1_2"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:45_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E3:45_1_2_1_1_1_2"/><b>then </b>
the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></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<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B 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<sub>1</sub></font> :  <a href="card_1.html#K1">Card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="triang_1.html#E2:45_1_2"><i><font color="Green">E15</font></i></a></i>;<br/><b>hence </b><a NAME="E4:45_1_2_1_1_1_2"/>
the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="triang_1.html#E5:45_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="triang_1.html#E2:45_1_2_1_1_1_2"><i><font color="Green">E21</font></i></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_2_1"/>
the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:45_1_2"/><b>then </b><i><font color="Green">E21</font></i>: 
the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R6">=</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_1_2_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = the <a href="triang_1.html#U2" target="_self">FacesAssign</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>7</sub></font> = the <a href="triang_1.html#U2" target="_self">FacesAssign</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> as   <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">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="triang_1.html#K4" target="_self">FuncsSeq</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="triang_1.html#E8:45_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/><a NAME="E4:45_1_2_2"/><i><font color="Green">E22</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <a href="triang_1.html#K6" target="_self">NatEmbSeq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <a href="triang_1.html#K6" target="_self">NatEmbSeq</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_1_2_2_1"/>
<font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="triang_1.html#NM5">Face</a> of <font color="Maroon">c<sub>5</sub></font> ;<br/><a NAME="E3:45_1_2_2_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="triang_1.html#K4" target="_self">FuncsSeq</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="triang_1.html#K4" target="_self">FuncsSeq</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><a NAME="E4:45_1_2_2_1"/><b>then </b><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default">the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default">the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="triang_1.html#D5" target="_self">Def5</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:45_1_2_2_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:45_1_2_2_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E4:45_1_2_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_1_2_2_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="triang_1.html#NM4">Symplex</a> of <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="triang_1.html#NM4">Symplex</a> of <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <b>by </b><i><a class="txt" href="triang_1.html#E7:45_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="triang_1.html#E1:45_1_2_2_1_1"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E4:45_1_2_2_1_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="triang_1.html#K7" target="_self">face</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="triang_1.html#K7" target="_self">face</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E8:45_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="triang_1.html#E6:45_1_2_2_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="triang_1.html#E8:45_1_2_2_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="triang_1.html#E1:45_1_2_2_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="triang_1.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E5:45_1_2_2_1_1"/>
<font color="Maroon">c<sub>13</sub></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<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B 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<sub>1</sub></font> :  <a href="card_1.html#K1">Card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 <b>by </b><i><a class="txt" href="triang_1.html#E2:45_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="triang_1.html#E1:45_1_2_2_1_1"><i><font color="Green">E26</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>15</sub></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<sub>1</sub></font><b> such that </b><br/><a NAME="E7:45_1_2_2_1_1"/><i><font color="Green">E28</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<sub>1</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><a NAME="E8:45_1_2_2_1_1"/>
(  <a href="triang_1.html#K7" target="_self">face</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="triang_1.html#K7" target="_self">face</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E3:45_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="triang_1.html#E6:45_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="triang_1.html#E8:45_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="triang_1.html#E1:45_1_2_2_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="triang_1.html#E7:45_1_2_2_1_1"><i><font color="Green">E28</font></i></a></i>;<br/><b>hence </b><a NAME="E9:45_1_2_2_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="triang_1.html#E4:45_1_2_2_1_1"><i><font color="Green">E27</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:45_1_2_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="triang_1.html#E6:45_1_2_2_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="triang_1.html#E8:45_1_2_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:45_1_2_2"/>
the <a href="triang_1.html#U2" target="_self">FacesAssign</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="triang_1.html#U2" target="_self">FacesAssign</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="triang_1.html#E4:45_1_2_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:45_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="triang_1.html#E8:45_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
