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<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>assume </b><a NAME="E1:7"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> is   <a href="finseq_1.html#NM1">FinSequence</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> as   <a href="finseq_1.html#NM1">FinSequence</a> ;<br/>
<a NAME="E3:7"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T24">RELAT_1:24</a></i>;<br/>
<a NAME="E4:7"/><b>then </b><i><font color="Green">E29</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E5:7"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E6:7"/><b>then </b><i><font color="Green">E30</font></i>: 
( <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E7:7"/><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:7"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="card_2.html#T69">CARD_2:69</a></i>;<br/>
<a NAME="E9:7"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>
<a NAME="E10:7"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E11:7"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><b>hence </b><a NAME="E2:7_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:7_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_2"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><a NAME="E2:7_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> implies <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/><a NAME="E3:7_2"/>
( ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 2 ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 2 ) )
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E4:7_2"/>
( ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 2 ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1 ) )
 <b>by </b><i>, , <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:7"/>
( ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 2 ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1 ) )
 <b>by </b><i>, , , <a class="ref" href="nat_1.html#T27">NAT_1:27</a></i>;<br/>


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