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<b>let </b><font color="Maroon" title="c1">E</font>, <font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">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/>





<div><i><font color="Green" title="E43">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:61_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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="c3">A</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> <b> such that </b><br/><a NAME="E3:61_1"/><i><font color="Green" title="E44">A2</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="ref" href="flang_1.html#D1" target="_self" title="FLANG_1:def.1">Def1</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:61_1"/><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="txt" href="flang_1.html#E3:61_1"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="flang_1.html#T42" target="_self" title="FLANG_1:th.42">Th42</a></i>;<br/><a NAME="E6:61_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> 1
 <b>by </b><i><a class="txt" href="flang_1.html#E3:61_1"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="flang_1.html#T26" target="_self" title="FLANG_1:th.26">Th26</a></i>;<br/><a NAME="E7:61_1"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="flang_1.html#E5:61_1"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="flang_1.html#T41" target="_self" title="FLANG_1:th.41">Th41</a></i>;<br/><b>hence </b><a NAME="E8:61_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a> 
 <b>by </b><i><a class="txt" href="flang_1.html#E3:61_1"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="flang_1.html#T42" target="_self" title="FLANG_1:th.42">Th42</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:61_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a> </span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> <b> such that </b><br/><a NAME="E3:61_2"/><i><font color="Green" title="E44">A4</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a>  &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="ref" href="flang_1.html#D1" target="_self" title="FLANG_1:def.1">Def1</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:61_2"/><i><font color="Green" title="E45">A5</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="txt" href="flang_1.html#E3:61_2"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="flang_1.html#T42" target="_self" title="FLANG_1:th.42">Th42</a></i>;<br/><a NAME="E6:61_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> 1
 <b>by </b><i><a class="txt" href="flang_1.html#E3:61_2"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="flang_1.html#T26" target="_self" title="FLANG_1:th.26">Th26</a></i>;<br/><a NAME="E7:61_2"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="flang_1.html#E5:61_2"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="flang_1.html#T41" target="_self" title="FLANG_1:th.41">Th41</a></i>;<br/><b>hence </b><a NAME="E8:61_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a> 
 <b>by </b><i><a class="txt" href="flang_1.html#E3:61_2"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="flang_1.html#T42" target="_self" title="FLANG_1:th.42">Th42</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:61"/>
( ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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="c3">A</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a> </span>)</span> ) implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a>  )
 <b>by </b><i><a class="txt" href="flang_1.html#E1:61"><i><font color="Green" title="E43">A1</font></i></a></i>;<br/>


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