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

<b>let </b><font color="Maroon">T1</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> , <font color="Maroon">T2</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">E130</font></i>: 
the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">T1</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">E131</font></i>: 
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 <font color="Maroon">T1</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">A</font></span>)</span> where <font color="Olive">A</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">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> </span>}</span> 
 <b>and </b><br/><a NAME="E3:45_1_2"/><i><font color="Green">E132</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a><br/> for <font color="Olive">f</font> being  <a href="triang_1.html#NM5">Face</a> of <font color="Olive">n</font><br/> for <font color="Olive">s</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">T1</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> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">T1</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/> <a href="triang_1.html#K7" target="_self">face</a> <font color="Olive">s</font>,<font color="Olive">f</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">f</font>
 <b>and </b><br/><a NAME="E4:45_1_2"/><i><font color="Green">E133</font></i>: 
the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">T2</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">E134</font></i>: 
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 <font color="Maroon">T2</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">A</font></span>)</span> where <font color="Olive">A</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">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> </span>}</span> 
 <b>and </b><br/><a NAME="E6:45_1_2"/><i><font color="Green">E135</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a><br/> for <font color="Olive">f</font> being  <a href="triang_1.html#NM5">Face</a> of <font color="Olive">n</font><br/> for <font color="Olive">s</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">T2</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> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">T2</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/> <a href="triang_1.html#K7" target="_self">face</a> <font color="Olive">s</font>,<font color="Olive">f</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">f</font>
 ;<br/>

<a NAME="E7:45_1_2"/><i><font color="Green">E136</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="numbers.html#K5">NAT</a>  holds <br/>the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_2_1"/>
<font color="Maroon">x</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">x</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_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">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_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_1_1"/>
<font color="Maroon">n</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">T1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">T2</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><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">n</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">E137</font></i>: 
<font color="Maroon">n</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">T1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">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="Maroon">n</font> </span>}</span> 
 <b>by </b><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">T1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">T2</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>
<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">T1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:45_1_2"/><b>then </b><i><font color="Green">E138</font></i>: 
the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">T1</font> <a href="pboole.html#R6">=</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">T2</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">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_1_2_2"/>
<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>reconsider </b><font color="Maroon">F1n</font> = the <a href="triang_1.html#U2" target="_self">FacesAssign</a> of <font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>, <font color="Maroon">F2n</font> = the <a href="triang_1.html#U2" target="_self">FacesAssign</a> of <font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</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">n</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">T1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <b>by </b><i>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/><a NAME="E4:45_1_2_2"/><i><font color="Green">E139</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F1n</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">n</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F2n</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">n</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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_1_2_2_1"/>
<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">n</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="triang_1.html#NM5">Face</a> of <font color="Maroon">n</font> ;<br/><a NAME="E3:45_1_2_2_1"/>
( <font color="Maroon">F1n</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</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">T1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">F2n</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</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">T1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 ;<br/><a NAME="E4:45_1_2_2_1"/><b>then </b><i><font color="Green">E140</font></i>: 
( <font color="Maroon">F1n</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</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">T1</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">the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F2n</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</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">T1</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">the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">T1</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">F1nx</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">E142</font></i>: 
( <font color="Maroon">F1nx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1n</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F1nx</font> <a href="hidden.html#R1">=</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">T1</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> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1nx</font> <a href="tarski.html#R1">c=</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">T1</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/><b>consider </b><font color="Maroon">F2nx</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">E143</font></i>: 
( <font color="Maroon">F2nx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2n</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F2nx</font> <a href="hidden.html#R1">=</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">T1</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> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F2nx</font> <a href="tarski.html#R1">c=</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">T1</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/><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">y</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">E144</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="triang_1.html#U1" target="_self">SkeletonSeq</a> of <font color="Maroon">T1</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>then reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="triang_1.html#NM4">Symplex</a> of <font color="Maroon">T1</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ;<br/><b>reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">y</font> as   <a href="triang_1.html#NM4">Symplex</a> of <font color="Maroon">T2</font>,<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="E4:45_1_2_2_1_1"/><i><font color="Green">E145</font></i>: 
( <font color="Maroon">F1nx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a>  <a href="triang_1.html#K7" target="_self">face</a> <font color="Maroon">y1</font>,<font color="Maroon">x1</font> &amp; <font color="Maroon">F2nx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a>  <a href="triang_1.html#K7" target="_self">face</a> <font color="Maroon">y2</font>,<font color="Maroon">x1</font> )
 <b>by </b><i>, , , , </i>;<br/><a NAME="E5:45_1_2_2_1_1"/>
<font color="Maroon">y1</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>, </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="E7:45_1_2_2_1_1"/><i><font color="Green">E146</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">y1</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/><a NAME="E8:45_1_2_2_1_1"/>
(  <a href="triang_1.html#K7" target="_self">face</a> <font color="Maroon">y1</font>,<font color="Maroon">x1</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">x1</font> &amp;  <a href="triang_1.html#K7" target="_self">face</a> <font color="Maroon">y2</font>,<font color="Maroon">x1</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">x1</font> )
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E9:45_1_2_2_1_1"/>
<font color="Maroon">F1nx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2nx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:45_1_2_2_1"/>
<font color="Maroon">F1n</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2n</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <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">T1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="triang_1.html#U2" target="_self">FacesAssign</a> of <font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <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">T1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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