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<div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c2">f</font> =  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">F</font></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">F</font> ;<br/>
<a NAME="E2:24_1_1"/>
 for <font color="Olive" title="b1">J</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">J</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">J</font>)</span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">J</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</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"> ( <font color="Maroon" title="c3">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> implies  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">J</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">J</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1_1_1"/>
<font color="Maroon" title="c3">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f</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="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">J</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">J</font>)</span>)</span></span><br/>

<a NAME="E2:24_1_1_1"/><span class="kw">then </span>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E3:24_1_1_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">J</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">J</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" onmouseover="rs('nat_1/T2')" onmouseout="rh()">NAT_1:2</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>
<a NAME="E3:24_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c2">f</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> 
 
;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c2">f</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"> <font color="Maroon" title="c2">f</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> </span><br/>

<span class="kw">thus </span><a NAME="E4:24_1_1"/>
<font color="Maroon" title="c2">f</font> is  <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1_1"><span class="lab"><font color="Green" title="E16">A1</font></span></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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