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

<b>let </b><font color="Maroon" title="c2">T1</font>, <font color="Maroon" title="c3">T2</font> be  <a href="triang_1.html#V2" title="TRIANG_1:attr.2">strict</a> <a href="triang_1.html#V3" title="TRIANG_1:attr.3">lower_non-empty</a>  <a href="triang_1.html#L1" title="TRIANG_1:struct.1">TriangStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:45_1_2"/><i><font color="Green" title="E14">A59</font></i>: 
the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>and </b><br/><a NAME="E2:45_1_2"/><i><font color="Green" title="E15">A60</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 holds <br/>the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b2">A</font></span>)</span> where <font color="Olive" title="b2">A</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> </span>}</span> 
 <b>and </b><br/><a NAME="E3:45_1_2"/><i><font color="Green" title="E16">A61</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">f</font> being  <a href="triang_1.html#NM5" title="TRIANG_1:NM.5">Face</a> of <font color="Olive" title="b1">n</font><br/> for <font color="Olive" title="b3">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>  st <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>for <font color="Olive" title="b4">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font>  st  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s</font> holds <br/> <a href="triang_1.html#K7" title="TRIANG_1:func.7">face</a> <font color="Olive" title="b3">s</font>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b4">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">f</font>
 <b>and </b><br/><a NAME="E4:45_1_2"/><i><font color="Green" title="E17">A62</font></i>: 
the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c3">T2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>and </b><br/><a NAME="E5:45_1_2"/><i><font color="Green" title="E18">A63</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 holds <br/>the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c3">T2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b2">A</font></span>)</span> where <font color="Olive" title="b2">A</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> </span>}</span> 
 <b>and </b><br/><a NAME="E6:45_1_2"/><i><font color="Green" title="E19">A64</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">f</font> being  <a href="triang_1.html#NM5" title="TRIANG_1:NM.5">Face</a> of <font color="Olive" title="b1">n</font><br/> for <font color="Olive" title="b3">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c3">T2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>  st <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c3">T2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>for <font color="Olive" title="b4">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font>  st  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s</font> holds <br/> <a href="triang_1.html#K7" title="TRIANG_1:func.7">face</a> <font color="Olive" title="b3">s</font>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b4">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">f</font>
 ;<br/>

<a NAME="E7:45_1_2"/><i><font color="Green" title="E20">A65</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/>the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c3">T2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</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" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_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" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_1_1"/>
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c3">T2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E1:45_1_2"><i><font color="Green" title="E14">A59</font></i></a>, <a class="txt" href="triang_1.html#E4:45_1_2"><i><font color="Green" title="E17">A62</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" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:45_1_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E21">A66</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><a NAME="E3:45_1_2_1_1_1_2"/><b>then </b>
the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">s</font></span>)</span> where <font color="Olive" title="b1">s</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="triang_1.html#E2:45_1_2"><i><font color="Green" title="E15">A60</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" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c3">T2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E5:45_1_2"><i><font color="Green" title="E18">A63</font></i></a>, <a class="txt" href="triang_1.html#E2:45_1_2_1_1_1_2"><i><font color="Green" title="E21">A66</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" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c3">T2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:45_1_2"/><b>then </b><i><font color="Green" title="E21">A67</font></i>: 
the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c3">T2</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">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" title="c4">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:45_1_2_2"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c4">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c6">F1n</font> = the <a href="triang_1.html#U2" title="TRIANG_1:sel.2">FacesAssign</a> of <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>, <font color="Maroon" title="c7">F2n</font> = the <a href="triang_1.html#U2" title="TRIANG_1:sel.2">FacesAssign</a> of <font color="Maroon" title="c3">T2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="triang_1.html#K6" title="TRIANG_1:func.6">NatEmbSeq</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>,<span class="p1">(<span class="default"><a href="triang_1.html#K4" title="TRIANG_1:func.4">FuncsSeq</a> the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <b>by </b><i><a class="txt" href="triang_1.html#E8:45_1_2"><i><font color="Green" title="E21">A67</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/><a NAME="E4:45_1_2_2"/><i><font color="Green" title="E22">A68</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">F1n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="triang_1.html#K6" title="TRIANG_1:func.6">NatEmbSeq</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">F2n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="triang_1.html#K6" title="TRIANG_1:func.6">NatEmbSeq</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">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" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:45_1_2_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="triang_1.html#K6" title="TRIANG_1:func.6">NatEmbSeq</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c9">x1</font> = <font color="Maroon" title="c8">x</font> as   <a href="triang_1.html#NM5" title="TRIANG_1:NM.5">Face</a> of <font color="Maroon" title="c5">n</font> ;<br/><a NAME="E3:45_1_2_2_1"/>
( <font color="Maroon" title="c6">F1n</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="triang_1.html#K4" title="TRIANG_1:func.4">FuncsSeq</a> the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c7">F2n</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="triang_1.html#K4" title="TRIANG_1:func.4">FuncsSeq</a> the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> )
 ;<br/><a NAME="E4:45_1_2_2_1"/><b>then </b><i><font color="Green" title="E23">A69</font></i>: 
( <font color="Maroon" title="c6">F1n</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default">the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> &amp; <font color="Maroon" title="c7">F2n</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default">the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="triang_1.html#D5" target="_self" title="TRIANG_1:def.5">Def5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">F1nx</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:45_1_2_2_1"/><i><font color="Green" title="E24">A70</font></i>: 
( <font color="Maroon" title="c10">F1nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F1n</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x1</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">F1nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">F1nx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">F2nx</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:45_1_2_2_1"/><i><font color="Green" title="E25">A71</font></i>: 
( <font color="Maroon" title="c11">F2nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F2n</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x1</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">F2nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">F2nx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E4:45_1_2_2_1"><i><font color="Green" title="E23">A69</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">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" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:45_1_2_2_1_1"/><i><font color="Green" title="E26">A72</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="triang_1.html#U1" title="TRIANG_1:sel.1">SkeletonSeq</a> of <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c13">y1</font> = <font color="Maroon" title="c12">y</font> as   <a href="triang_1.html#NM4" title="TRIANG_1:NM.4">Symplex</a> of <font color="Maroon" title="c2">T1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ;<br/><b>reconsider </b><font color="Maroon" title="c14">y2</font> = <font color="Maroon" title="c12">y</font> as   <a href="triang_1.html#NM4" title="TRIANG_1:NM.4">Symplex</a> of <font color="Maroon" title="c3">T2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <b>by </b><i><a class="txt" href="triang_1.html#E7:45_1_2"><i><font color="Green" title="E20">A65</font></i></a>, <a class="txt" href="triang_1.html#E1:45_1_2_2_1_1"><i><font color="Green" title="E26">A72</font></i></a></i>;<br/><a NAME="E4:45_1_2_2_1_1"/><i><font color="Green" title="E27">A73</font></i>: 
( <font color="Maroon" title="c10">F1nx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="triang_1.html#K7" title="TRIANG_1:func.7">face</a> <font color="Maroon" title="c13">y1</font>,<font color="Maroon" title="c9">x1</font> &amp; <font color="Maroon" title="c11">F2nx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="triang_1.html#K7" title="TRIANG_1:func.7">face</a> <font color="Maroon" title="c14">y2</font>,<font color="Maroon" title="c9">x1</font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E8:45_1_2"><i><font color="Green" title="E21">A67</font></i></a>, <a class="txt" href="triang_1.html#E6:45_1_2_2_1"><i><font color="Green" title="E24">A70</font></i></a>, <a class="txt" href="triang_1.html#E8:45_1_2_2_1"><i><font color="Green" title="E25">A71</font></i></a>, <a class="txt" href="triang_1.html#E1:45_1_2_2_1_1"><i><font color="Green" title="E26">A72</font></i></a>, <a class="ref" href="triang_1.html#D10" target="_self" title="TRIANG_1:def.10">Def10</a></i>;<br/><a NAME="E5:45_1_2_2_1_1"/>
<font color="Maroon" title="c13">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">s</font></span>)</span> where <font color="Olive" title="b1">s</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 </span>}</span> 
 <b>by </b><i><a class="txt" href="triang_1.html#E2:45_1_2"><i><font color="Green" title="E15">A60</font></i></a>, <a class="txt" href="triang_1.html#E1:45_1_2_2_1_1"><i><font color="Green" title="E26">A72</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">A</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="triang_1.html#M1" title="TRIANG_1:mode.1">Element</a> of  <a href="triang_1.html#K3" title="TRIANG_1:func.3">symplexes</a> <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E7:45_1_2_2_1_1"/><i><font color="Green" title="E28">A74</font></i>: 
(  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c15">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c15">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><a NAME="E8:45_1_2_2_1_1"/>
(  <a href="triang_1.html#K7" title="TRIANG_1:func.7">face</a> <font color="Maroon" title="c13">y1</font>,<font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c15">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">x1</font> &amp;  <a href="triang_1.html#K7" title="TRIANG_1:func.7">face</a> <font color="Maroon" title="c14">y2</font>,<font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c15">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">x1</font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E3:45_1_2"><i><font color="Green" title="E16">A61</font></i></a>, <a class="txt" href="triang_1.html#E6:45_1_2"><i><font color="Green" title="E19">A64</font></i></a>, <a class="txt" href="triang_1.html#E8:45_1_2"><i><font color="Green" title="E21">A67</font></i></a>, <a class="txt" href="triang_1.html#E1:45_1_2_2_1_1"><i><font color="Green" title="E26">A72</font></i></a>, <a class="txt" href="triang_1.html#E7:45_1_2_2_1_1"><i><font color="Green" title="E28">A74</font></i></a></i>;<br/><b>hence </b><a NAME="E9:45_1_2_2_1_1"/>
<font color="Maroon" title="c10">F1nx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F2nx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E4:45_1_2_2_1_1"><i><font color="Green" title="E27">A73</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:45_1_2_2_1"/>
<font color="Maroon" title="c6">F1n</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F2n</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E6:45_1_2_2_1"><i><font color="Green" title="E24">A70</font></i></a>, <a class="txt" href="triang_1.html#E8:45_1_2_2_1"><i><font color="Green" title="E25">A71</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">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" title="TRIANG_1:sel.2">FacesAssign</a> of <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="triang_1.html#U2" title="TRIANG_1:sel.2">FacesAssign</a> of <font color="Maroon" title="c3">T2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E4:45_1_2_2"><i><font color="Green" title="E22">A68</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:45_1_2"/>
<font color="Maroon" title="c2">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T2</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E8:45_1_2"><i><font color="Green" title="E21">A67</font></i></a>, <a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/>


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