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<span class="kw">thus </span><a NAME="E1:19_1_1"/>
( <font color="Maroon" title="c1">A</font> is  <a href="matroid0.html#NV6" title="MATROID0:NV.6">finite-membered</a>  implies  for <font color="Olive" title="b1">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Olive" title="b1">B</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</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">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Olive" title="b1">B</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  ) implies <font color="Maroon" title="c1">A</font> is  <a href="matroid0.html#NV6" title="MATROID0:NV.6">finite-membered</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:19_1_1_1"/>
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds  <font color="Olive" title="b1">a</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lexbfs.html#D1" title="LEXBFS:def.1">LEXBFS:def 1</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">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Olive" title="b1">B</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/>

<span class="kw">hence </span><a NAME="E2:19_1_1_1"/>
 for <font color="Olive" title="b1">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Olive" title="b1">B</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ; <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">assume </span><a NAME="E2:19_1_1"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
 for <font color="Olive" title="b1">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Olive" title="b1">B</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ; <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="c1">A</font> is  <a href="matroid0.html#NV6" title="MATROID0:NV.6">finite-membered</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lexbfs.html#D1" title="LEXBFS:def.1">LEXBFS:def 1</a> <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">a</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/>

<a NAME="E3:19_1_1"/>
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
;<br/>
<span class="kw">hence </span><a NAME="E4:19_1_1"/>
<font color="Maroon" title="c2">a</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_1"><span class="lab"><font color="Green" title="E7">A1</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/>


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