<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <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> being   <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/> for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">J</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st not <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">J</font> holds <br/> <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b4">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">x</font></span>)</span>,<font color="Olive" title="b4">J</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be    <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>; <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">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">J</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st not <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">J</font> holds <br/> <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b3">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">x</font></span>)</span>,<font color="Olive" title="b3">J</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">x</font>, <font color="Olive" title="b2">J</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st not <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">J</font> holds <br/> <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b2">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">x</font></span>)</span>,<font color="Olive" title="b2">J</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">J</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"> ( not <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font> implies  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c5">J</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:20"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
not <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c5">J</font></span><br/>

<span class="kw">thus </span><a NAME="E2:20"/>
 <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c5">J</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="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c5">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">J</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">z</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="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">J</font> or <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c5">J</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_1"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">J</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="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c5">J</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:20_1"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font>
 <span class="kw">and </span><br/><a NAME="E4:20_1"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E5:20_1"/><span class="lab"><font color="Green" title="E18">A4</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hallmar1.html#D1" target="_self" title="HALLMAR1:def.1">Def1</a></span>;<br/>
<a NAME="E6:20_1"/><span class="lab"><font color="Green" title="E19">A5</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="hallmar1.html#D2" target="_self" title="HALLMAR1:def.2">Def2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_1_1"/>
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_1_1_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</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="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c5">J</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:20_1_1_1"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c5">J</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E3:20_1"><span class="lab"><font color="Green" title="E16">A2</font></span></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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_1_1_2"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</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="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c5">J</font></span><br/><div class="add"><a NAME="E2:20_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="ref" href="hallmar1.html#D2" target="_self" title="HALLMAR1:def.2">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:20_1_1_2"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c5">J</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E5:20_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E6:20_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="ref" href="hallmar1.html#D1" target="_self" title="HALLMAR1: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></div><span class="kw">end;</span></div>

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

<span class="kw">let </span><font color="Maroon" title="c6">z</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="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c5">J</font> or <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">J</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:20"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c5">J</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="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">J</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:20"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font>
 <span class="kw">and </span><br/><a NAME="E6:20"/><span class="lab"><font color="Green" title="E17">A7</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">x</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:20"/><span class="lab"><font color="Green" title="E18">A8</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hallmar1.html#D1" target="_self" title="HALLMAR1:def.1">Def1</a></span>;<br/>
<a NAME="E8:20"/><span class="lab"><font color="Green" title="E19">A9</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="ref" href="hallmar1.html#D2" target="_self" title="HALLMAR1:def.2">Def2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_2"/>
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_1"><span class="kw">suppose </span></a><a NAME="E1:20_2_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</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="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">J</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:20_2_1"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">J</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E16">A6</font></span></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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_2"><span class="kw">suppose </span></a><a NAME="E1:20_2_2"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</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="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">J</font></span><br/><div class="add"><a NAME="E2:20_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="hallmar1.html#K2" title="HALLMAR1:func.2">Cut</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="ref" href="hallmar1.html#D2" target="_self" title="HALLMAR1:def.2">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:20_2_2"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hallmar1.html#K1" title="HALLMAR1:func.1">union</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">J</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E7:20"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="ref" href="hallmar1.html#D1" target="_self" title="HALLMAR1: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></div><span class="kw">end;</span></div>

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