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

<span class="kw">reconsider </span><font color="Maroon" title="c4">V</font> =  <a href="simplex0.html#K4" title="SIMPLEX0:func.4">Vertices</a> <font color="Maroon" title="c2">KX</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">KX</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D5" target="_self" onmouseover="rs('simplex0/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">PP</font> =  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">TOP</font> =  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>)</span>)</span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">D2</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">KX</font> &amp; <font color="Maroon" title="c3">P</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Olive" title="b1">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> );<br/>
<a NAME="E2:124_1_1"/>
 <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="simplex0.html#E29"><span class="lab"><font color="Green" title="E21">Lm5</font></span></a></span>;<br/>
<a NAME="E3:124_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A1</font></span>: 
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c4">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T82" onmouseover="rs('zfmisc_1/T82')" onmouseout="rh()">ZFMISC_1:82</a></span>;<br/>
<a NAME="E4:124_1_1"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>)</span>)</span> holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c4">V</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="124_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>)</span>)</span> implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c4">V</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:124_1_1_1"/><span class="lab"><font color="Green" title="E49">A3</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c4">V</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">X</font> = <font color="Maroon" title="c7">x</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:124_1_1_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a></span>;<br/>
<a NAME="E3:124_1_1_1"/>
<font color="Maroon" title="c8">X</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:124_1_1_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">S</font> being   <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a>  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">KX</font><span class="kw"> such that </span><br/><a NAME="E5:124_1_1_1"/><span class="lab"><font color="Green" title="E50">A4</font></span>: 
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c9">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D20" target="_self" onmouseover="rs('simplex0/D20')" onmouseout="rh()">Def20</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">S</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c4">V</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">S</font>] )</span><br/>

<a NAME="E6:124_1_1_1"/>
<font color="Maroon" title="c9">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="124_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">S</font> or <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:124_1_1_1_1"/><span class="lab"><font color="Green" title="E51">A5</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">S</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">Y</font> = <font color="Maroon" title="c10">y</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">KX</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:124_1_1_1_1"><span class="lab"><font color="Green" title="E51">A5</font></span></a></span>;<br/>
<a NAME="E3:124_1_1_1_1"/>
<font color="Maroon" title="c11">Y</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:124_1_1_1_1"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="tops_2.html#D1" onmouseover="rs('tops_2/D1')" onmouseout="rh()">TOPS_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:124_1_1_1_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:124_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c4">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:124_1_1"><span class="lab"><font color="Green" title="E47">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:124_1_1_1"/>
( <font color="Maroon" title="c9">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c4">V</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">S</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:124_1_1_1"><span class="lab"><font color="Green" title="E50">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c7">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E6:124_1_1"/><span class="lab"><font color="Green" title="E49">A6</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>)</span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" onmouseover="rs('funct_2/S1')" onmouseout="rh()">FUNCT_2:sch 1</a>(<a class="txt" href="#E4:124_1_1"><span class="lab"><font color="Green" title="E48">A2</font></span></a>);<br/></span>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="124_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c9">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x2</font> implies <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:124_1_1_2"/><span class="lab"><font color="Green" title="E50">A7</font></span>: 
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 <span class="kw">and </span><br/><a NAME="E2:124_1_1_2"/><span class="lab"><font color="Green" title="E51">A8</font></span>: 
( <font color="Maroon" title="c9">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x2</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font></span><br/><a NAME="E3:124_1_1_2"/><span class="lab"><font color="Green" title="E52">A9</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x2</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:124_1_1"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="txt" href="#E2:124_1_1_2"><span class="lab"><font color="Green" title="E51">A8</font></span></a></span>;<br/><a NAME="E4:124_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:124_1_1"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="txt" href="#E1:124_1_1_2"><span class="lab"><font color="Green" title="E50">A7</font></span></a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c8">x1</font> = 
<font color="Maroon" title="c3">P</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x1</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c9">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:124_1_1_2"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="txt" href="#E3:124_1_1_2"><span class="lab"><font color="Green" title="E52">A9</font></span></a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E8:124_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A10</font></span>: 
<font color="Maroon" title="c7">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>;<br/>
<a NAME="E9:124_1_1"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>)</span>)</span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c4">V</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E10:124_1_1"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>)</span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:124_1_1"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="ref" href="card_1.html#T10" onmouseover="rs('card_1/T10')" onmouseout="rh()">CARD_1:10</a></span>;<br/>
<a NAME="E11:124_1_1"/><span class="kw">then </span>
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>)</span>)</span> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E12:124_1_1"/>
 <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">KX</font>) is  <a href="simplex0.html#V2" title="SIMPLEX0:attr.2">finite-vertices</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T20" target="_self" onmouseover="rs('simplex0/T20')" onmouseout="rh()">Th20</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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