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<b>let </b><font color="Maroon" title="c1">E</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span>;<br/>



<b>assume </b><a NAME="E1:63"/><i><font color="Green" title="E44">A1</font></i>: 
 <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<a NAME="E2:63"/><i><font color="Green" title="E45">A2</font></i>: 
 <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a> 
 
<b>by </b><i><a class="ref" href="flang_1.html#T49" target="_self" title="FLANG_1:th.49">Th49</a></i>;<br/>
<a NAME="E3:63"/><i><font color="Green" title="E46">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a> </span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c2">A</font> implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a>  ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a> </span>)</span> implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a>  ) )
 
<b>by </b><i><a class="ref" href="flang_1.html#T53" target="_self" title="FLANG_1:th.53">Th53</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:63_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:63_1"/><i><font color="Green" title="E47">A4</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="flang_1.html#T42" target="_self" title="FLANG_1:th.42">Th42</a></i>;<br/><div><i><font color="Green" title="E48">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1_1"/>
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:63_1_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="flang_1.html#K4" title="FLANG_1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font></span>)</span></span><a href="flang_1.html#K4" title="FLANG_1:func.4">}</a></span>
 <b>by </b><i><a class="txt" href="flang_1.html#E3:63_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="flang_1.html#T25" target="_self" title="FLANG_1:th.25">Th25</a></i>;<br/><a NAME="E3:63_1_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:63_1_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a> </span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="flang_1.html#E1:63"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="flang_1.html#E2:63"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="flang_1.html#T16" target="_self" title="FLANG_1:th.16">Th16</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E49">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1_2"/>
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><b>then consider </b><font color="Maroon" title="c5">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:63_1_2"/><i><font color="Green" title="E50">A7</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>thus </b><a NAME="E4:63_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a> </span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="flang_1.html#E3:63_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="flang_1.html#E3:63_1_2"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="flang_1.html#T51" target="_self" title="FLANG_1:th.51">Th51</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1_3"/>
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><b>then consider </b><font color="Maroon" title="c5">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:63_1_3"/><i><font color="Green" title="E50">A8</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>thus </b><a NAME="E4:63_1_3"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a> </span>)</span>
 <b>by </b><i><a class="txt" href="flang_1.html#E3:63_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="flang_1.html#E3:63_1_3"><i><font color="Green" title="E50">A8</font></i></a>, <a class="ref" href="flang_1.html#T51" target="_self" title="FLANG_1:th.51">Th51</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:63_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a> </span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="flang_1.html#E4:63_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="flang_1.html#E5:63_1"><i><font color="Green" title="E49">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:63"/><b>then </b>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a>  implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a> </span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c2">A</font> ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a>  implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a> </span>)</span> ) )
 
;<br/>
<b>hence </b><a NAME="E6:63"/>
( <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a> </span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="flang_1.html#E3:63"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


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