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<div class="add">

<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">P</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/>  for <font color="Olive" title="b2">K1</font>, <font color="Olive" title="b3">K2</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font>  st  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">K1</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Olive" title="b2">K1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">K2</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Olive" title="b3">K2</font> #) holds <br/> <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">K1</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="Olive" title="b3">K2</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">  for <font color="Olive" title="b1">K1</font>, <font color="Olive" title="b2">K2</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font>  st  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">K1</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Olive" title="b1">K1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">K2</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Olive" title="b2">K2</font> #) holds <br/> <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c2">P</font>,<font color="Olive" title="b1">K1</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="c2">P</font>,<font color="Olive" title="b2">K2</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">K1</font>, <font color="Maroon" title="c4">K2</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"> (  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K1</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">K1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">K2</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">K2</font> #) implies  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">K1</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="c2">P</font>,<font color="Maroon" title="c4">K2</font>) )</span><br/>





<span class="kw">assume </span><a NAME="E1:132"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K1</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">K1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">K2</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">K2</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> (<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">K1</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="c2">P</font>,<font color="Maroon" title="c4">K2</font>)</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">P1</font> =  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">K1</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">P2</font> =  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c4">K2</font>);<br/>
<a NAME="E2:132"/><span class="lab"><font color="Green" title="E52">A2</font></span>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">P</font>,<font color="Maroon" title="c3">K1</font>)</span>)</span> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">P</font>,<font color="Maroon" title="c4">K2</font>)</span>)</span> )
 
<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="E3:132"/><span class="lab"><font color="Green" title="E53">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="c2">P</font>,<font color="Maroon" title="c3">K1</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="c2">P</font>,<font color="Maroon" title="c4">K2</font>)</span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="132_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="c2">P</font>,<font color="Maroon" title="c3">K1</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="c2">P</font>,<font color="Maroon" title="c4">K2</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:132_1"/><span class="lab"><font color="Green" title="E54">A4</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="c2">P</font>,<font color="Maroon" title="c3">K1</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="c2">P</font>,<font color="Maroon" title="c4">K2</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="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="c2">P</font>,<font color="Maroon" title="c3">K1</font>)</span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">A1</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="c2">P</font>,<font color="Maroon" title="c4">K2</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:132"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E2:132"><span class="lab"><font color="Green" title="E52">A2</font></span></a></span>;<br/>
<a NAME="E4:132_1"/>
<font color="Maroon" title="c8">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="txt" href="#E1:132_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a></span>;<br/>
<span class="kw">then </span><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">K1</font><span class="kw"> such that </span><br/><a NAME="E6:132_1"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">S1</font> = <font color="Maroon" title="c10">S</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c4">K2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:132"><span class="lab"><font color="Green" title="E51">A1</font></span></a></span>;<br/>
<a NAME="E8:132_1"/>
<font color="Maroon" title="c10">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">K1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T14" target="_self" onmouseover="rs('simplex0/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<a NAME="E9:132_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">S1</font> is  <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:132"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="simplex0.html#T14" target="_self" onmouseover="rs('simplex0/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<a NAME="E10:132_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">A1</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="#E6:132_1"><span class="lab"><font color="Green" title="E55">A5</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="E11:132_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="c2">P</font>,<font color="Maroon" title="c4">K2</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="E4:132"/>
 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="c2">P</font>,<font color="Maroon" title="c4">K2</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="c2">P</font>,<font color="Maroon" title="c3">K1</font>)</span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="132_2"><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="c2">P</font>,<font color="Maroon" title="c4">K2</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="c2">P</font>,<font color="Maroon" title="c3">K1</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:132_2"/><span class="lab"><font color="Green" title="E54">A6</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="c2">P</font>,<font color="Maroon" title="c4">K2</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="c2">P</font>,<font color="Maroon" title="c3">K1</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="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="c2">P</font>,<font color="Maroon" title="c4">K2</font>)</span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">A1</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="c2">P</font>,<font color="Maroon" title="c3">K1</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:132"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E2:132"><span class="lab"><font color="Green" title="E52">A2</font></span></a></span>;<br/>
<a NAME="E4:132_2"/>
<font color="Maroon" title="c8">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="txt" href="#E1:132_2"><span class="lab"><font color="Green" title="E54">A6</font></span></a></span>;<br/>
<span class="kw">then </span><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="c4">K2</font><span class="kw"> such that </span><br/><a NAME="E6:132_2"/><span class="lab"><font color="Green" title="E55">A7</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">S1</font> = <font color="Maroon" title="c10">S</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c3">K1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:132"><span class="lab"><font color="Green" title="E51">A1</font></span></a></span>;<br/>
<a NAME="E8:132_2"/>
<font color="Maroon" title="c10">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="c4">K2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T14" target="_self" onmouseover="rs('simplex0/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<a NAME="E9:132_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">S1</font> is  <a href="simplex0.html#NV3" title="SIMPLEX0:NV.3">simplex-like</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:132"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="simplex0.html#T14" target="_self" onmouseover="rs('simplex0/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<a NAME="E10:132_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">A1</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="#E6:132_2"><span class="lab"><font color="Green" title="E55">A7</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="E11:132_2"/>
<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="c2">P</font>,<font color="Maroon" title="c3">K1</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>
<span class="kw">hence </span><a NAME="E5:132"/>
 <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">K1</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="c2">P</font>,<font color="Maroon" title="c4">K2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:132"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E2:132"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E3:132"><span class="lab"><font color="Green" title="E53">A3</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>
