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<span class="kw">let </span><font color="Maroon" title="c1">E</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  holds <br/> ( <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="flang_1.html#K9" title="FLANG_1:func.9">Lex</a> <font color="Maroon" title="c1">E</font> iff  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="flang_1.html#K9" title="FLANG_1:func.9">Lex</a> <font color="Maroon" title="c1">E</font> iff  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>



<span class="kw">thus </span><a NAME="E1:10"/>
( <font color="Maroon" title="c2">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="flang_1.html#K9" title="FLANG_1:func.9">Lex</a> <font color="Maroon" title="c1">E</font> implies  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c2">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="flang_1.html#K9" title="FLANG_1:func.9">Lex</a> <font color="Maroon" title="c1">E</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:10_1"/>
<font color="Maroon" title="c2">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="flang_1.html#K9" title="FLANG_1:func.9">Lex</a> <font color="Maroon" title="c1">E</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<a NAME="E2:10_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <br/>( <font color="Olive" title="b1">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="flang_1.html#K3" title="FLANG_1:func.3">&lt;%</a><span class="default"><font color="Olive" title="b1">e</font></span><a href="flang_1.html#K3" title="FLANG_1:func.3">%&gt;</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="flang_1.html#D4" onmouseover="rs('flang_1/D4')" onmouseout="rh()">FLANG_1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:10_1"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#D4" onmouseover="rs('afinsq_1/D4')" onmouseout="rh()">AFINSQ_1:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E2:10"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="flang_1.html#K9" title="FLANG_1:func.9">Lex</a> <font color="Maroon" title="c1">E</font></span><br/>

<a NAME="E3:10"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <br/>( <span class="p1"><a href="flang_1.html#K3" title="FLANG_1:func.3">&lt;%</a><span class="default"><font color="Olive" title="b1">e</font></span><a href="flang_1.html#K3" title="FLANG_1:func.3">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#T4" target="_self" onmouseover="rs('fsm_3/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:10"/>
<font color="Maroon" title="c2">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="flang_1.html#K9" title="FLANG_1:func.9">Lex</a> <font color="Maroon" title="c1">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="flang_1.html#D4" onmouseover="rs('flang_1/D4')" onmouseout="rh()">FLANG_1:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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