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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">F1</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">A1</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>  st ( for <font color="Olive" title="b3">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">F1</font> holds <br/><font color="Olive" title="b2">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b3">k</font> ) &amp; ( for <font color="Olive" title="b3">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st not <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">F1</font> holds <br/><font color="Olive" title="b2">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) holds <br/>( <font color="Olive" title="b2">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Olive" title="b2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_3.html#K8" title="PROB_3:func.8">Union</a> <font color="Olive" title="b1">F1</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">F1</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</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">A1</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>  st ( for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">F1</font> holds <br/><font color="Olive" title="b1">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b2">k</font> ) &amp; ( for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st not <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">F1</font> holds <br/><font color="Olive" title="b1">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) holds <br/>( <font color="Olive" title="b1">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Olive" title="b1">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_3.html#K8" title="PROB_3:func.8">Union</a> <font color="Maroon" title="c2">F1</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">A1</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</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">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">F1</font> holds <br/><font color="Maroon" title="c3">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">k</font> ) &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st not <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">F1</font> holds <br/><font color="Maroon" title="c3">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) implies ( <font color="Maroon" title="c3">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_3.html#K8" title="PROB_3:func.8">Union</a> <font color="Maroon" title="c2">F1</font> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:54"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">F1</font> holds <br/><font color="Maroon" title="c3">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">k</font>
 <span class="kw">and </span><br/><a NAME="E2:54"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st not <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">F1</font> holds <br/><font color="Maroon" title="c3">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_3.html#K8" title="PROB_3:func.8">Union</a> <font color="Maroon" title="c2">F1</font> )</span><br/>

<span class="kw">thus </span><a NAME="E3:54"/>
<font color="Maroon" title="c3">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="prob_3.html#T1" target="_self" title="PROB_3:th.1">Th1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_3.html#K8" title="PROB_3:func.8">Union</a> <font color="Maroon" title="c2">F1</font></span><br/>

<span class="kw">thus </span><a NAME="E4:54"/>
 <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_3.html#K8" title="PROB_3:func.8">Union</a> <font color="Maroon" title="c2">F1</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:54_1"/>
 <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c3">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_3.html#K8" title="PROB_3:func.8">Union</a> <font color="Maroon" title="c2">F1</font>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="prob_3.html#K8" title="PROB_3:func.8">Union</a> <font color="Maroon" title="c2">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c3">A1</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c3">A1</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_3.html#K8" title="PROB_3:func.8">Union</a> <font color="Maroon" title="c2">F1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:54_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c3">A1</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_3.html#K8" title="PROB_3:func.8">Union</a> <font color="Maroon" title="c2">F1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:54_1_1"/><span class="lab"><font color="Green" title="E49">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T25" title="PROB_1:th.25">PROB_1:25</a></span>;<br/>
<a NAME="E4:54_1_1"/>
( <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">F1</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c5">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E3:54_1_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:54_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_3.html#K8" title="PROB_3:func.8">Union</a> <font color="Maroon" title="c2">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T54" target="_self" title="PROB_3:th.54">Th54</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/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_3.html#K8" title="PROB_3:func.8">Union</a> <font color="Maroon" title="c2">F1</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c3">A1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:54_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_3.html#K8" title="PROB_3:func.8">Union</a> <font color="Maroon" title="c2">F1</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c3">A1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:54_1"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
( <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">F1</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c5">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T54" target="_self" title="PROB_3:th.54">Th54</a></span>;<br/>
<a NAME="E5:54_1"/>
( <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E4:54_1"><span class="lab"><font color="Green" title="E49">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:54_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c3">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T25" title="PROB_1:th.25">PROB_1:25</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/>


</div><span class="kw">end;</span></div>


</div>
