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<span class="kw">let </span><font color="Maroon" title="c1">E</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">A</font>, <font color="Olive" title="b2">B</font>, <font color="Olive" title="b3">C</font> being  <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#K2" title="CATALAN2:func.2">^omega</a> </span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Olive" title="b3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Olive" title="b3">C</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">C</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#K2" title="CATALAN2:func.2">^omega</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c4">C</font> <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="c3">B</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c4">C</font></span>)</span></span><br/>







<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">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"> ( ( <font color="Maroon" title="c5">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#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c4">C</font> implies <font color="Maroon" title="c5">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="c3">B</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c4">C</font></span>)</span> ) &amp; ( <font color="Maroon" title="c5">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="c3">B</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c4">C</font></span>)</span> implies <font color="Maroon" title="c5">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#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c4">C</font> ) )</span><br/><span class="kw">thus </span><a NAME="E1:25_1"/>
( <font color="Maroon" title="c5">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#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c4">C</font> implies <font color="Maroon" title="c5">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="c3">B</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c4">C</font></span>)</span> )
  <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="c5">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="c3">B</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c4">C</font></span>)</span> implies <font color="Maroon" title="c5">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#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c4">C</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:25_1_1"/>
<font color="Maroon" title="c5">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#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c4">C</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="c5">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="c3">B</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c4">C</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">ab</font>, <font color="Maroon" title="c7">c</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#K2" title="CATALAN2:func.2">^omega</a> <span class="kw"> such that </span><br/><a NAME="E3:25_1_1"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c6">ab</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> <font color="Maroon" title="c3">B</font>
 <span class="kw">and </span><br/><a NAME="E4:25_1_1"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
( <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">ab</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="flang_1.html#D1" target="_self" title="FLANG_1:def.1">Def1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">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#K2" title="CATALAN2:func.2">^omega</a> <span class="kw"> such that </span><br/><a NAME="E6:25_1_1"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E7:25_1_1"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
( <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c6">ab</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c9">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_1_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="flang_1.html#D1" target="_self" title="FLANG_1:def.1">Def1</a></span>;<br/>
<a NAME="E8:25_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">c</font></span>)</span> &amp; <font color="Maroon" title="c9">b</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c4">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_1_1"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E7:25_1_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="flang_1.html#D1" target="_self" title="FLANG_1:def.1">Def1</a>, <a class="ref" href="afinsq_1.html#T30" title="AFINSQ_1:th.30">AFINSQ_1:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:25_1_1"/>
<font color="Maroon" title="c5">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="c3">B</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c4">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="flang_1.html#D1" target="_self" title="FLANG_1:def.1">Def1</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">assume </span><a NAME="E2:25_1"/>
<font color="Maroon" title="c5">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="c3">B</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c4">C</font></span>)</span>
 ; <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="c5">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#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c4">C</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">bc</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#K2" title="CATALAN2:func.2">^omega</a> <span class="kw"> such that </span><br/><a NAME="E4:25_1"/><span class="lab"><font color="Green" title="E19">A5</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E5:25_1"/><span class="lab"><font color="Green" title="E20">A6</font></span>: 
<font color="Maroon" title="c7">bc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E6:25_1"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">bc</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="flang_1.html#D1" target="_self" title="FLANG_1:def.1">Def1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">c</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#K2" title="CATALAN2:func.2">^omega</a> <span class="kw"> such that </span><br/><a NAME="E8:25_1"/><span class="lab"><font color="Green" title="E22">A8</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">and </span><br/><a NAME="E9:25_1"/><span class="lab"><font color="Green" title="E23">A9</font></span>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E10:25_1"/><span class="lab"><font color="Green" title="E24">A10</font></span>: 
<font color="Maroon" title="c7">bc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c9">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="flang_1.html#D1" target="_self" title="FLANG_1:def.1">Def1</a></span>;<br/><a NAME="E11:25_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c6">a</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c8">b</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> <font color="Maroon" title="c3">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E6:25_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E8:25_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E10:25_1"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="flang_1.html#D1" target="_self" title="FLANG_1:def.1">Def1</a>, <a class="ref" href="afinsq_1.html#T30" title="AFINSQ_1:th.30">AFINSQ_1:30</a></span>;<br/><span class="kw">hence </span><a NAME="E12:25_1"/>
<font color="Maroon" title="c5">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#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:25_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="ref" href="flang_1.html#D1" target="_self" title="FLANG_1:def.1">Def1</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">hence </span><a NAME="E2:25"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c4">C</font> <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="c3">B</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c4">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</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>
