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<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> holds <br/> ( <font color="Olive" title="b1">K</font> is  <a href="simplex1.html#V2" title="SIMPLEX1:attr.2">simplex-join-closed</a>  iff  for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">K</font>  st  not <font color="Olive" title="b2">A</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &amp;  not <font color="Olive" title="b3">B</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &amp;  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b2">A</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Olive" title="b3">B</font></span>)</span> holds <br/><font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">B</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"> ( <font color="Maroon" title="c2">K</font> is  <a href="simplex1.html#V2" title="SIMPLEX1:attr.2">simplex-join-closed</a>  iff  for <font color="Olive" title="b1">A</font>, <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  not <font color="Olive" title="b1">A</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &amp;  not <font color="Olive" title="b2">B</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &amp;  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</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="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</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> holds <br/><font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font> )</span><br/>



<div><span class="kw">hereby </span> <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>, <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  not <font color="Olive" title="b1">A</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &amp;  not <font color="Olive" title="b2">B</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &amp;  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</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="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</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> holds <br/><font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font> ) implies <font color="Maroon" title="c2">K</font> is  <a href="simplex1.html#V2" title="SIMPLEX1:attr.2">simplex-join-closed</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:47_1"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<font color="Maroon" title="c2">K</font> is  <a href="simplex1.html#V2" title="SIMPLEX1:attr.2">simplex-join-closed</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>, <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  not <font color="Olive" title="b1">A</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &amp;  not <font color="Olive" title="b2">B</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &amp;  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</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="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</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> holds <br/> not <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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"> (  not <font color="Maroon" title="c3">A</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &amp;  not <font color="Maroon" title="c4">B</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &amp;  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c4">B</font></span>)</span> implies  not <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">B</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:47_1"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
(  not <font color="Maroon" title="c3">A</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &amp;  not <font color="Maroon" title="c4">B</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  )
 <span class="kw">and </span><br/><a NAME="E3:47_1"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
 <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c4">B</font></span>)</span>
 ; <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="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">B</font></span><br/><a NAME="E4:47_1"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
<span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p2">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p2">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E2:47_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="simplex1.html#D8" target="_self" title="SIMPLEX1:def.8">Def8</a></span>;<br/><span class="kw">assume </span><a NAME="E5:47_1"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E6:47_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A5</font></span>: 
( <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">A</font> or <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">B</font> )
 ;<br/><a NAME="E7:47_1"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
( <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:47_1"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:47_1"/><span class="lab"><font color="Green" title="E37">A8</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E11:47_1"/>
(  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c3">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="Maroon" title="c3">A</font></span>)</span> &amp;  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c4">B</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="Maroon" title="c4">B</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin2.html#T5" title="RLAFFIN2:th.5">RLAFFIN2:5</a></span>;<br/><a NAME="E12:47_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A9</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p2">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p2">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:47_1"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E10:47_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_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="E1:47_1_1"/>
( <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c3">A</font> or <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c4">B</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:47_1"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E7:47_1"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:47_1_1_1"/>
<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:47_1_1_1"/><span class="kw">then </span>
 <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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin2.html#T7" title="RLAFFIN2:th.7">RLAFFIN2:7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:47_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E9:47_1"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E12:47_1"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</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="47_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:47_1_1_2"/>
<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:47_1_1_2"/><span class="kw">then </span>
 <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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin2.html#T7" title="RLAFFIN2:th.7">RLAFFIN2:7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:47_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E10:47_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E12:47_1"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</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></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:47"/><span class="lab"><font color="Green" title="E30">A10</font></span>: 
 for <font color="Olive" title="b1">A</font>, <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  not <font color="Olive" title="b1">A</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &amp;  not <font color="Olive" title="b2">B</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &amp;  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</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="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</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> holds <br/><font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</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="c2">K</font> is  <a href="simplex1.html#V2" title="SIMPLEX1:attr.2">simplex-join-closed</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">K</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="simplex1.html#D8" target="_self" title="SIMPLEX1:def.8">SIMPLEX1:def 8</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="c3">A</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &amp;  not <font color="Maroon" title="c4">B</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  implies <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p2">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p2">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:47"/><span class="lab"><font color="Green" title="E31">A11</font></span>: 
 not <font color="Maroon" title="c3">A</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a> 
 <span class="kw">and </span><br/><a NAME="E4:47"/><span class="lab"><font color="Green" title="E32">A12</font></span>: 
 not <font color="Maroon" title="c4">B</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"> <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p2">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p2">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span></span><br/>

<a NAME="E5:47"/><span class="lab"><font color="Green" title="E33">A13</font></span>: 
<span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p2">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p2">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c4">B</font></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p2">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p2">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> or <font color="Maroon" title="c5">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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> )</span><br/>

<a NAME="E1:47_2"/><span class="lab"><font color="Green" title="E34">A14</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/>
<span class="kw">assume </span><a NAME="E2:47_2"/><span class="lab"><font color="Green" title="E35">A15</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p2">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p2">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</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="c5">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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span></span><br/>

<a NAME="E3:47_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">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="c3">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:47_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c3">A</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin2.html#T8" title="RLAFFIN2:th.8">RLAFFIN2:8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">Ia</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:47_2"/><span class="lab"><font color="Green" title="E36">A16</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Ia</font>
 <span class="kw">and </span><br/><a NAME="E7:47_2"/><span class="lab"><font color="Green" title="E37">A17</font></span>: 
<font color="Maroon" title="c6">Ia</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="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c3">A</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">a</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E9:47_2"/><span class="lab"><font color="Green" title="E38">A18</font></span>: 
<font color="Maroon" title="c6">Ia</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">and </span><br/><a NAME="E10:47_2"/><span class="lab"><font color="Green" title="E39">A19</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47_2"><span class="lab"><font color="Green" title="E37">A17</font></span></a></span>;<br/>
<a NAME="E11:47_2"/>
<font color="Maroon" title="c5">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="c4">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_2"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E12:47_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Olive" title="b1">b</font></span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c4">B</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin2.html#T8" title="RLAFFIN2:th.8">RLAFFIN2:8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">Ib</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E14:47_2"/><span class="lab"><font color="Green" title="E40">A20</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Ib</font>
 <span class="kw">and </span><br/><a NAME="E15:47_2"/><span class="lab"><font color="Green" title="E41">A21</font></span>: 
<font color="Maroon" title="c8">Ib</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="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Olive" title="b1">b</font></span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c4">B</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">b</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E17:47_2"/><span class="lab"><font color="Green" title="E42">A22</font></span>: 
<font color="Maroon" title="c8">Ib</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c9">b</font>
 <span class="kw">and </span><br/><a NAME="E18:47_2"/><span class="lab"><font color="Green" title="E43">A23</font></span>: 
<font color="Maroon" title="c9">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:47_2"><span class="lab"><font color="Green" title="E41">A21</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">a1</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c11">b1</font> = <font color="Maroon" title="c9">b</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:47_2"><span class="lab"><font color="Green" title="E39">A19</font></span></a>, <a class="txt" href="#E18:47_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E20:47_2"/>
<font color="Maroon" title="c3">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="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E31">A11</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="E21:47_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">a1</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="#E1:47_2"><span class="lab"><font color="Green" title="E34">A14</font></span></a>, <a class="txt" href="#E10:47_2"><span class="lab"><font color="Green" title="E39">A19</font></span></a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></span>;<br/>
<a NAME="E22:47_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A24</font></span>: 
 not <font color="Maroon" title="c10">a1</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/>
<a NAME="E23:47_2"/>
<font color="Maroon" title="c4">B</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="#E4:47"><span class="lab"><font color="Green" title="E32">A12</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="E24:47_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">b1</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="#E1:47_2"><span class="lab"><font color="Green" title="E34">A14</font></span></a>, <a class="txt" href="#E18:47_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></span>;<br/>
<a NAME="E25:47_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A25</font></span>: 
 not <font color="Maroon" title="c11">b1</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/>
<a NAME="E26:47_2"/>
 <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c10">a1</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <span class="p1">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c11">b1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:47_2"><span class="lab"><font color="Green" title="E36">A16</font></span></a>, <a class="txt" href="#E9:47_2"><span class="lab"><font color="Green" title="E38">A18</font></span></a>, <a class="txt" href="#E14:47_2"><span class="lab"><font color="Green" title="E40">A20</font></span></a>, <a class="txt" href="#E17:47_2"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E27:47_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E30">A10</font></span></a>, <a class="txt" href="#E22:47_2"><span class="lab"><font color="Green" title="E44">A24</font></span></a>, <a class="txt" href="#E25:47_2"><span class="lab"><font color="Green" title="E45">A25</font></span></a></span>;<br/>
<a NAME="E28:47_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:47_2"><span class="lab"><font color="Green" title="E39">A19</font></span></a>, <a class="txt" href="#E18:47_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/>
<a NAME="E29:47_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A26</font></span>: 
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c7">a</font> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T3" title="RLAFFIN1:th.3">RLAFFIN1:3</a></span>;<br/>
<a NAME="E30:47_2"/>
<font color="Maroon" title="c5">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="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:47_2"><span class="lab"><font color="Green" title="E36">A16</font></span></a>, <a class="txt" href="#E9:47_2"><span class="lab"><font color="Green" title="E38">A18</font></span></a>, <a class="ref" href="rlaffin2.html#D1" title="RLAFFIN2:def.1">RLAFFIN2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E31:47_2"/>
<font color="Maroon" title="c5">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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:47_2"><span class="lab"><font color="Green" title="E46">A26</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>
<a NAME="E6:47"/>
(  <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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></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="Maroon" title="c3">A</font></span>)</span> &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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></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="Maroon" title="c4">B</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T3" title="RLAFFIN1:th.3">RLAFFIN1:3</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E7:47"/><span class="kw">then </span>
 <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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p2">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p2">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:47"/>
<span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p2">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <span class="p2">(<span class="default"><a href="simplex1.html#K1" title="SIMPLEX1:func.1">@</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47"><span class="lab"><font color="Green" title="E33">A13</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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>
