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

<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">K</font> being   <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a>  <a href="simplex1.html#NM1" title="SIMPLEX1:NM.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">V</font><br/>  for <font color="Olive" title="b2">SF</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Olive" title="b1">K</font>  st <font color="Olive" title="b2">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cohsp_1.html#K2" title="COHSP_1:func.2">Sub_of_Fin</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Olive" title="b1">K</font> holds <br/> <a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <font color="Olive" title="b2">SF</font> is    <a href="simplex1.html#M1" title="SIMPLEX1:mode.1">SubdivisionStr</a> of <font color="Olive" title="b1">K</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">K</font> be   <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a>  <a href="simplex1.html#NM1" title="SIMPLEX1:NM.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">SF</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">K</font>  st <font color="Olive" title="b1">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cohsp_1.html#K2" title="COHSP_1:func.2">Sub_of_Fin</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">K</font> holds <br/> <a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <font color="Olive" title="b1">SF</font> is    <a href="simplex1.html#M1" title="SIMPLEX1:mode.1">SubdivisionStr</a> of <font color="Maroon" title="c2">K</font></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">TOP</font> =  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">K</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">SF</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">K</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="c4">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cohsp_1.html#K2" title="COHSP_1:func.2">Sub_of_Fin</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">K</font> implies  <a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <font color="Maroon" title="c4">SF</font> is    <a href="simplex1.html#M1" title="SIMPLEX1:mode.1">SubdivisionStr</a> of <font color="Maroon" title="c2">K</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:19"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c4">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cohsp_1.html#K2" title="COHSP_1:func.2">Sub_of_Fin</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <font color="Maroon" title="c4">SF</font> is    <a href="simplex1.html#M1" title="SIMPLEX1:mode.1">SubdivisionStr</a> of <font color="Maroon" title="c2">K</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">C</font> =  <a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <font color="Maroon" title="c4">SF</font>;<br/>
<a NAME="E2:19"/>
(  <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> <font color="Maroon" title="c4">SF</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">K</font> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D9" title="SIMPLEX0:def.9">SIMPLEX0:def 9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">C</font> =  <a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <font color="Maroon" title="c4">SF</font> as   <a href="simplex1.html#NM1" title="SIMPLEX1:NM.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D9" title="SIMPLEX0:def.9">SIMPLEX0:def 9</a></span>;<br/>
<a NAME="E4:19"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
 <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <font color="Maroon" title="c2">K</font> is  <a href="classes1.html#V1" title="CLASSES1:attr.1">subset-closed</a> 
 
;<br/>
<a NAME="E5:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A3</font></span>: 
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">SF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="simplex0.html#T7" title="SIMPLEX0:th.7">SIMPLEX0:7</a></span>;<br/>
<a NAME="E6:19"/>
<font color="Maroon" title="c6">C</font> is    <a href="simplex1.html#M1" title="SIMPLEX1:mode.1">SubdivisionStr</a> of <font color="Maroon" title="c2">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:19_1"/>
<span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">K</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c6">C</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="simplex1.html#D4" target="_self" title="SIMPLEX1:def.4">SIMPLEX1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">C</font>  st  not <font color="Olive" title="b1">A</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  holds <br/> ex <font color="Olive" title="b2">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">K</font> st <br/>(  not <font color="Olive" title="b2">B</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b2">B</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_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="firebrick">:: 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="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> <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">K</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c6">C</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:19_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">K</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span>
 ; <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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c6">C</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E3:19_1_1"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
 not <font color="Maroon" title="c8">S</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a> 
 <span class="kw">and </span><br/><a NAME="E4:19_1_1"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c8">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex1.html#D3" target="_self" title="SIMPLEX1:def.3">Def3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">S1</font> =  <a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c8">S</font> as   non  <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="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a></span>;<br/>
<a NAME="E6:19_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">L</font></span>)</span> where <font color="Olive" title="b1">L</font> is   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c9">S1</font> : <font color="Olive" title="b1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c1">V</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="convex3.html#T5" title="CONVEX3:th.5">CONVEX3:5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">L</font> being   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c9">S1</font><span class="kw"> such that </span><br/><a NAME="E8:19_1_1"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c10">L</font> &amp; <font color="Maroon" title="c10">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c1">V</font> )
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">Carr</font> =  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c10">L</font> as   non  <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="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="convex1.html#T21" title="CONVEX1:th.21">CONVEX1:21</a></span>;<br/>
<a NAME="E10:19_1_1"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
<font color="Maroon" title="c11">Carr</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D6" title="RLVECT_2:def.6">RLVECT_2:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">Carr1</font> = <font color="Maroon" title="c11">Carr</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E12:19_1_1"/>
<font color="Maroon" title="c8">S</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">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/>
<a NAME="E13:19_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">Carr1</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">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E10:19_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></span>;<br/>
<a NAME="E14:19_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">Carr1</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">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="cohsp_1.html#D3" title="COHSP_1:def.3">COHSP_1:def 3</a></span>;<br/>
<a NAME="E15:19_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A8</font></span>: 
 not <font color="Maroon" title="c12">Carr1</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">LC</font> = <font color="Maroon" title="c10">L</font> as    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c11">Carr</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D6" title="RLVECT_2:def.6">RLVECT_2:def 6</a></span>;<br/>
<a NAME="E17:19_1_1"/>
<font color="Maroon" title="c13">LC</font> is  <a href="convex1.html#V2" title="CONVEX1:attr.2">convex</a> 
 
;<br/>
<a NAME="E18:19_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">M</font></span>)</span> where <font color="Olive" title="b1">M</font> is   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c11">Carr</font> : <font color="Olive" title="b1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c1">V</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:19_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a></span>;<br/>
<a NAME="E19:19_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A9</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c11">Carr</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="convex3.html#T5" title="CONVEX3:th.5">CONVEX3:5</a></span>;<br/>
<a NAME="E20:19_1_1"/>
<font color="Maroon" title="c11">Carr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c12">Carr1</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E21:19_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c6">C</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:19_1_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E19:19_1_1"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="simplex1.html#D3" target="_self" title="SIMPLEX1:def.3">Def3</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>

<span class="kw">let </span><font color="Maroon" title="c7">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">A</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  implies  ex <font color="Olive" title="b1">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">K</font> st <br/>(  not <font color="Olive" title="b1">B</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">B</font></span>)</span> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">B</font> = <font color="Maroon" title="c7">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">K</font> ;<br/>
<span class="kw">assume </span><a NAME="E3:19_1"/>
 not <font color="Maroon" title="c7">A</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">K</font> st <br/>(  not <font color="Olive" title="b1">B</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">B</font></span>)</span> )</span><br/>

<a NAME="E4:19_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</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">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/>
<a NAME="E5:19_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A10</font></span>: 
 not <font color="Maroon" title="c8">B</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/>
<a NAME="E6:19_1"/>
 <a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c8">B</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E7:19_1"/>
 ex <font color="Olive" title="b1">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">K</font> st <br/>(  not <font color="Olive" title="b1">B</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b1">B</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19_1"><span class="lab"><font color="Green" title="E17">A10</font></span></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>
<span class="kw">hence </span><a NAME="E7:19"/>
 <a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <font color="Maroon" title="c4">SF</font> is    <a href="simplex1.html#M1" title="SIMPLEX1:mode.1">SubdivisionStr</a> of <font color="Maroon" title="c2">K</font>
 ; <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>
