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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">A</font>, <font color="Olive" title="b2">B</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">A</font> is  <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  &amp;  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Olive" title="b2">B</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Olive" title="b2">B</font> is  <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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"> ( <font color="Maroon" title="c2">A</font> is  <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  &amp;  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">B</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">B</font> implies <font color="Maroon" title="c3">B</font> is  <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:105"/><span class="lab"><font color="Green" title="E77">A1</font></span>: 
( <font color="Maroon" title="c2">A</font> is  <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  &amp;  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">B</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">B</font> )
 ; <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">B</font> is  <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a> </span><br/>

<a NAME="E2:105"/>
 <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">Ib</font> being   <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</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"> such that </span><br/><a NAME="E4:105"/>
 <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Ib</font>
 <span class="kw">and </span><br/><a NAME="E5:105"/><span class="lab"><font color="Green" title="E78">A2</font></span>: 
<font color="Maroon" title="c4">Ib</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">and </span><br/><a NAME="E6:105"/><span class="lab"><font color="Green" title="E79">A3</font></span>: 
 <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">Ib</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T60" target="_self" title="RLAFFIN1:th.60">Th60</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">IB</font> = <font color="Maroon" title="c4">Ib</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</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="#E5:105"><span class="lab"><font color="Green" title="E78">A2</font></span></a></span>;<br/>
<a NAME="E8:105"/><span class="lab"><font color="Green" title="E80">A4</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">IB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:105"><span class="lab"><font color="Green" title="E79">A3</font></span></a>, <a class="ref" href="rlaffin1.html#T79" target="_self" title="RLAFFIN1:th.79">Th79</a></span>;<br/>
<a NAME="E9:105"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">IB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="txt" href="#E6:105"><span class="lab"><font color="Green" title="E79">A3</font></span></a>, <a class="ref" href="rlaffin1.html#T79" target="_self" title="RLAFFIN1:th.79">Th79</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:105"/>
<font color="Maroon" title="c3">B</font> is  <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:105"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="txt" href="#E8:105"><span class="lab"><font color="Green" title="E80">A4</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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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