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<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">  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 <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b2">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_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="Olive" title="b2">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</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" 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">P</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_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="Olive" title="b1">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</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" 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"> ( <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_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="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</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">PX</font> =  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c3">KX</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">PP</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="E1:127"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T58" onmouseover="rs('relat_1/T58')" onmouseout="rh()">RELAT_1:58</a>, <a class="ref" href="relat_1.html#T74" onmouseover="rs('relat_1/T74')" onmouseout="rh()">RELAT_1:74</a></span>;<br/>
<a NAME="E2:127"/><span class="lab"><font color="Green" title="E49">A2</font></span>: 
(  <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> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<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> &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" onmouseover="rs('simplex0/D20')" onmouseout="rh()">Def20</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:127"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_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" 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">  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</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="E4:127"/><span class="lab"><font color="Green" title="E51">A4</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 <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c3">KX</font>)</span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="127_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> ; <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="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 <span class="p1">(<span class="default"><a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c3">KX</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:127_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" 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> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c3">KX</font>)</span>)</span></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#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">B</font> = <font color="Maroon" title="c8">A</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> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c3">KX</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:127"><span class="lab"><font color="Green" title="E49">A2</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">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="E5:127_1"/><span class="lab"><font color="Green" title="E52">A5</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#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c10">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/>
<a NAME="E6:127_1"/><span class="lab"><font color="Green" title="E53">A6</font></span>: 
<font color="Maroon" title="c10">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="127_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</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="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">P</font></span>)</span> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:127_1_1"/><span class="lab"><font color="Green" title="E54">A7</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">P</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"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">A</font> = <font color="Maroon" title="c11">x</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">KX</font> ;<br/>
<a NAME="E3:127_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1_1"><span class="lab"><font color="Green" title="E54">A7</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:127_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">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="tops_2.html#D1" onmouseover="rs('tops_2/D1')" onmouseout="rh()">TOPS_2:def 1</a></span>;<br/>
<a NAME="E5:127_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A8</font></span>: 
<font color="Maroon" title="c12">A</font> <a href="tarski.html#R2" title="TARSKI: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>
 
;<br/>
<a NAME="E6:127_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1_1"><span class="lab"><font color="Green" title="E54">A7</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E7:127_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">A</font> <a href="tarski.html#R2" title="TARSKI: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> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">P</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:127_1_1"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:127_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:127"><span class="lab"><font color="Green" title="E50">A3</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>
<a NAME="E7:127_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A9</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">P</font></span>)</span></span>)</span> <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">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">P</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>;<br/>
<font color="Maroon" title="c9">B</font> = 
<font color="Maroon" title="c4">P</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">P</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:127_1"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="ref" href="relat_1.html#T112" onmouseover="rs('relat_1/T112')" onmouseout="rh()">RELAT_1:112</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">P</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c2">Y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:127_1"><span class="lab"><font color="Green" title="E53">A6</font></span></a>, <a class="txt" href="#E7:127_1"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="ref" href="relat_1.html#T129" onmouseover="rs('relat_1/T129')" onmouseout="rh()">RELAT_1:129</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" onmouseover="rs('xboole_1/T16')" onmouseout="rh()">XBOOLE_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">S</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c10">S</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T112" onmouseover="rs('relat_1/T112')" onmouseout="rh()">RELAT_1:112</a></span>
;<br/>
<a NAME="E9:127_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">B</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" onmouseover="rs('simplex0/D20')" onmouseout="rh()">Def20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:127_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> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c3">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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:127"/>
( <font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> )
 
;<br/>
<a NAME="E6:127"/><span class="kw">then </span>
 <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c3">KX</font>) is    <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="txt" href="#E1:127"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="simplex0.html#T54" target="_self" onmouseover="rs('simplex0/T54')" onmouseout="rh()">Th54</a>, <a class="ref" href="relat_1.html#T60" onmouseover="rs('relat_1/T60')" onmouseout="rh()">RELAT_1:60</a></span>;<br/>
<a NAME="E7:127"/><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> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<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 <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" onmouseover="rs('simplex0/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:127"/>
 <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">Y</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="#E2:127"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E4:127"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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>
