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

<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">KX</font> being   <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">P</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b2">P</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Olive" title="b1">KX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> holds <br/> <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">|</a> <font color="Olive" title="b2">P</font></span>)</span>,<font color="Olive" title="b1">KX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> <font color="Olive" title="b2">P</font>,<font color="Olive" title="b1">KX</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">KX</font> be    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">P</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">KX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> holds <br/> <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">|</a> <font color="Olive" title="b1">P</font></span>)</span>,<font color="Maroon" title="c3">KX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> <font color="Olive" title="b1">P</font>,<font color="Maroon" title="c3">KX</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">P</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">P</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">KX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> implies  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">|</a> <font color="Maroon" title="c4">P</font></span>)</span>,<font color="Maroon" title="c3">KX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">KX</font> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">PK</font> =  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">KX</font>;<br/>
<a NAME="E1:135"/>
( <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">|</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">|</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">P</font></span>)</span></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">|</a> <font color="Maroon" title="c4">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T126" title="RELAT_1:th.126">RELAT_1:126</a>, <a class="ref" href="relat_1.html#T127" title="RELAT_1:th.127">RELAT_1:127</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">PY</font> =  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">|</a> <font color="Maroon" title="c4">P</font></span>)</span>,<font color="Maroon" title="c3">KX</font> as    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">KX</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T56" target="_self" title="SIMPLEX0:th.56">Th56</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E3:135"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c6">PY</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">KX</font> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">KX</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">KX</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D20" target="_self" title="SIMPLEX0:def.20">Def20</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:135"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
<font color="Maroon" title="c4">P</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">KX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">|</a> <font color="Maroon" title="c4">P</font></span>)</span>,<font color="Maroon" title="c3">KX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">KX</font></span><br/>

<a NAME="E5:135"/><span class="lab"><font color="Green" title="E54">A3</font></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="c4">P</font>,<font color="Maroon" title="c3">KX</font></span>)</span> <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="c6">PY</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="135_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">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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="c4">P</font>,<font color="Maroon" title="c3">KX</font></span>)</span> or <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 <font color="Maroon" title="c6">PY</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:135_1"/>
<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="c4">P</font>,<font color="Maroon" title="c3">KX</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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 <font color="Maroon" title="c6">PY</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">A</font> = <font color="Maroon" title="c7">x</font> as   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">KX</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<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="c3">KX</font><span class="kw"> such that </span><br/><a NAME="E4:135_1"/><span class="lab"><font color="Green" title="E55">A4</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</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" title="SIMPLEX0:def.20">Def20</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">A</font> = <font color="Maroon" title="c8">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">PY</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:135"><span class="lab"><font color="Green" title="E52">A1</font></span></a></span>;<br/>
<a NAME="E6:135_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="c3">KX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="135_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">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="c11">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="c3">KX</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:135_1_1"/><span class="lab"><font color="Green" title="E56">A5</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">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="c3">KX</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">y</font> = <font color="Maroon" title="c11">y</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">KX</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:135_1_1"><span class="lab"><font color="Green" title="E56">A5</font></span></a></span>;<br/>
<a NAME="E3:135_1_1"/>
<font color="Maroon" title="c12">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="#E1:135_1_1"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:135_1_1"/>
<font color="Maroon" title="c11">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="c3">KX</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:135_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">KX</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:135_1"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></span>;<br/>
<a NAME="E8:135_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:135"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E9:135_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">|</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c9">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:135_1"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="ref" href="funct_1.html#T126" title="FUNCT_1:th.126">FUNCT_1:126</a></span>;<br/>
<a NAME="E10:135_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">A</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D20" target="_self" title="SIMPLEX0:def.20">Def20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:135_1"/>
<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 <font color="Maroon" title="c6">PY</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:135"/>
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">PY</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 <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">KX</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D13" target="_self" title="SIMPLEX0:def.13">Def13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:135"/>
 <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">|</a> <font color="Maroon" title="c4">P</font></span>)</span>,<font color="Maroon" title="c3">KX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">KX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:135"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E5:135"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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