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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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Ks</font> being  <a href="simplex1.html#V2" title="SIMPLEX1:attr.2">simplex-join-closed</a>  <a href="simplex1.html#NM2" title="SIMPLEX1:NM.2">SimplicialComplex</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b2">As</font>, <font color="Olive" title="b3">Bs</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">Ks</font>  st not <font color="Olive" title="b2">As</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &amp; not <font color="Olive" title="b3">Bs</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">As</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</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="b3">Bs</font></span>)</span> holds <br/><font color="Olive" title="b2">As</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">Bs</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Ks</font> be   <a href="simplex1.html#V2" title="SIMPLEX1:attr.2">simplex-join-closed</a>  <a href="simplex1.html#NM2" title="SIMPLEX1:NM.2">SimplicialComplex</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">As</font>, <font color="Olive" title="b2">Bs</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Ks</font>  st not <font color="Olive" title="b1">As</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &amp; not <font color="Olive" title="b2">Bs</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">As</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</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">Bs</font></span>)</span> holds <br/><font color="Olive" title="b1">As</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Bs</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">As</font>, <font color="Maroon" title="c4">Bs</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Ks</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c3">As</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a>  &amp; not <font color="Maroon" title="c4">Bs</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">As</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</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">Bs</font></span>)</span> implies <font color="Maroon" title="c3">As</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Bs</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:49"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
not <font color="Maroon" title="c3">As</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a> 
 <span class="kw">and </span><br/><a NAME="E2:49"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
not <font color="Maroon" title="c4">Bs</font> is  <a href="matroid0.html#NV2" title="MATROID0:NV.2">dependent</a> 
 <span class="kw">and </span><br/><a NAME="E3:49"/><span class="lab"><font color="Green" title="E33">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">As</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</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">Bs</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">As</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Bs</font></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="E5:49"/><span class="lab"><font color="Green" title="E34">A4</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">As</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:49"/><span class="lab"><font color="Green" title="E35">A5</font></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="c4">Bs</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49"><span class="lab"><font color="Green" title="E33">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="E7:49"/>
<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">Bs</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <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">Ib</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:49"/><span class="lab"><font color="Green" title="E36">A6</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">Ib</font>
 <span class="kw">and </span><br/><a NAME="E10:49"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
<font color="Maroon" title="c6">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">Bs</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">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="E12:49"/><span class="lab"><font color="Green" title="E38">A8</font></span>: 
<font color="Maroon" title="c6">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="c7">b</font>
 <span class="kw">and </span><br/><a NAME="E13:49"/><span class="lab"><font color="Green" title="E39">A9</font></span>: 
<font color="Maroon" title="c7">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">Bs</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:49"><span class="lab"><font color="Green" title="E37">A7</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">b1</font> = <font color="Maroon" title="c7">b</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Ks</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:49"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E15:49"/>
<font color="Maroon" title="c3">As</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">Ks</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E16:49"/><span class="kw">then </span>
not <font color="Maroon" title="c2">Ks</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="ref" href="pencil_1.html#D4" title="PENCIL_1:def.4">PENCIL_1:def 4</a></span>;<br/>
<a NAME="E17:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A10</font></span>: 
not <font color="Maroon" title="c8">b1</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="#E2:49"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E13:49"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="matroid0.html#T1" title="MATROID0:th.1">MATROID0:1</a></span>;<br/>
<a NAME="E18:49"/>
 <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">As</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="c8">b1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E9:49"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="txt" href="#E12:49"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:49"/>
<font color="Maroon" title="c3">As</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Bs</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E13:49"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="txt" href="#E17:49"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="ref" href="simplex1.html#T25" target="_self" title="SIMPLEX1:th.25">Th25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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