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<a NAME="E1:83_1_1"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
 <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <font color="Maroon" title="c2">KD</font> is  <a href="classes1.html#V1" title="CLASSES1:attr.1">subset-closed</a> 
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:83_1_1"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c3">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="c2">KD</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:83_1_1_1"/>
( <font color="Maroon" title="c3">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c3">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:83_1_1_1_1"/>
<font color="Maroon" title="c3">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not  for <font color="Olive">b<sub>1</sub></font> being    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Maroon" title="c2">KD</font> holds  <font color="Olive">b<sub>1</sub></font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">x</font> as   <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">KD</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c5">S</font> =  <a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><a NAME="E3:83_1_1_1_1"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></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">KD</font>
 ;<br/><a NAME="E4:83_1_1_1_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">KD</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D9" target="_self" title="SIMPLEX0:def.9">Def9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">S</font> =  <a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="simplex0.html#NM4" title="SIMPLEX0:NM.4">SimplicialComplex</a> of <font color="Maroon" title="c1">D</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1_1_1_1"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="simplex0.html#D9" target="_self" title="SIMPLEX0:def.9">Def9</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c6">S</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">S</font> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Maroon" title="c2">KD</font> &amp;  not <font color="Maroon" title="c6">S</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  )</span><br/><a NAME="E6:83_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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="c2">KD</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/><a NAME="E7:83_1_1_1_1"/><span class="kw">then </span>
 <a href="simplex0.html#K2" title="SIMPLEX0:func.2">subset-closed_closure_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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="c2">KD</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1_1"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="simplex0.html#T3" target="_self" title="SIMPLEX0:th.3">Th3</a></span>;<br/><span class="kw">hence </span><a NAME="E8:83_1_1_1_1"/>
( <font color="Maroon" title="c6">S</font> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Maroon" title="c2">KD</font> &amp;  not <font color="Maroon" title="c6">S</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1_1_1_1"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="simplex0.html#D13" target="_self" title="SIMPLEX0:def.13">Def13</a></span>; <a class="txt" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:83_1_1_1_2"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
 not <font color="Maroon" title="c3">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not  for <font color="Olive">b<sub>1</sub></font> being    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Maroon" title="c2">KD</font> holds  <font color="Olive">b<sub>1</sub></font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> </span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c4">d</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">x</font>;<br/><a NAME="E2:83_1_1_1_2"/>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1_1_1_2"><span class="lab"><font color="Green" title="E36">A4</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">dd</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">KD</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c6">S</font> =  <a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">dd</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><a NAME="E4:83_1_1_1_2"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">dd</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></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">KD</font>
 ;<br/><a NAME="E5:83_1_1_1_2"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">KD</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D9" target="_self" title="SIMPLEX0:def.9">Def9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">S</font> =  <a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">dd</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="simplex0.html#NM4" title="SIMPLEX0:NM.4">SimplicialComplex</a> of <font color="Maroon" title="c1">D</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_1_1_1_2"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="simplex0.html#D9" target="_self" title="SIMPLEX0:def.9">Def9</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c7">S</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">S</font> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Maroon" title="c2">KD</font> &amp;  not <font color="Maroon" title="c7">S</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  )</span><br/><a NAME="E7:83_1_1_1_2"/>
<font color="Maroon" title="c5">dd</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1_1_1_2"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/><a NAME="E8:83_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">dd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KD</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1_1"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E3:83_1_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></span>;<br/><a NAME="E9:83_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">dd</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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="c2">KD</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/><a NAME="E10:83_1_1_1_2"/><span class="kw">then </span>
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KD</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1_1"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="simplex0.html#T3" target="_self" title="SIMPLEX0:th.3">Th3</a></span>;<br/><span class="kw">hence </span><a NAME="E11:83_1_1_1_2"/>
( <font color="Maroon" title="c7">S</font> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Maroon" title="c2">KD</font> &amp;  not <font color="Maroon" title="c7">S</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_1_1_1_2"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="simplex0.html#D13" target="_self" title="SIMPLEX0:def.13">Def13</a></span>; <a class="txt" 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></div><span class="kw">end;</span></div>

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