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<span class="kw">let </span><font color="Maroon" title="c1">X</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">SX</font> being    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Olive" title="b1">KX</font><br/>  for <font color="Olive" title="b3">P</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds   <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Olive" title="b3">P</font>,<font color="Olive" title="b2">SX</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="Olive" title="b3">P</font>,<font color="Olive" title="b1">KX</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">SX</font> being    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Maroon" title="c2">KX</font><br/>  for <font color="Olive" title="b2">P</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds   <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Olive" title="b2">P</font>,<font color="Olive" title="b1">SX</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="Olive" title="b2">P</font>,<font color="Maroon" title="c2">KX</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">SX</font> be    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</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">  for <font color="Olive" title="b1">P</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds   <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Olive" title="b1">P</font>,<font color="Maroon" title="c3">SX</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="Olive" title="b1">P</font>,<font color="Maroon" title="c2">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">  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">SX</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="c2">KX</font>)</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">PS</font> =  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">SX</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">PK</font> =  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">KX</font>);<br/>
<a NAME="E1:130"/><span class="lab"><font color="Green" title="E50">A1</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">SX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">KX</font>
 
<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/>
<a NAME="E2:130"/><span class="lab"><font color="Green" title="E51">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> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">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="c2">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">hence </span><a NAME="E3:130"/>
 <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">SX</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c2">KX</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:130"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="ref" href="simplex0.html#D20" target="_self" onmouseover="rs('simplex0/D20')" onmouseout="rh()">Def20</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="simplex0.html#D13" target="_self" onmouseover="rs('simplex0/D13')" onmouseout="rh()">SIMPLEX0:def 13</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">  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">SX</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="c2">KX</font>)</span>)</span></span><br/>

<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">SX</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> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">KX</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E4:130"/>
<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">SX</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> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">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">SX</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/>
<a NAME="E6:130"/>
 <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">SX</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">SX</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">then </span><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> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">KX</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:130"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="txt" href="#E2:130"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">SS</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">SX</font><span class="kw"> such that </span><br/><a NAME="E9:130"/><span class="lab"><font color="Green" title="E52">A3</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">SS</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="E10:130"/>
 <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">SX</font></span>)</span> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">KX</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:130"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T67" onmouseover="rs('zfmisc_1/T67')" onmouseout="rh()">ZFMISC_1:67</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">SK</font> = <font color="Maroon" title="c10">SS</font> as   <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="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">KX</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<a NAME="E12:130"/>
<font color="Maroon" title="c11">SK</font> is  <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="130_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">KX</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tops_2.html#D1" onmouseover="rs('tops_2/D1')" onmouseout="rh()">TOPS_2:def 1</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="c12">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">SK</font> or <font color="Maroon" title="c12">C</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:130_1"/><span class="lab"><font color="Green" title="E53">A4</font></span>: 
<font color="Maroon" title="c12">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">SK</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="c12">C</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">c</font> = <font color="Maroon" title="c12">C</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">SX</font> ;<br/>
<a NAME="E3:130_1"/>
<font color="Maroon" title="c13">c</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="#E1:130_1"><span class="lab"><font color="Green" title="E53">A4</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/>
<a NAME="E4:130_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A5</font></span>: 
<font color="Maroon" title="c13">c</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">SX</font>
 
;<br/>
<a NAME="E5:130_1"/>
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">SX</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>
 
<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="E6:130_1"/>
<font color="Maroon" title="c12">C</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="#E4:130_1"><span class="lab"><font color="Green" title="E54">A5</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="E13:130"/><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="txt" href="#E9:130"><span class="lab"><font color="Green" title="E52">A3</font></span></a>, <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="E14:130"/>
<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="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"> verum</span><br/>


</div>
