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

<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds   <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="finseq_3.html#K1" title="FINSEQ_3:func.1">-</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b1">A</font></span><br/><b>let </b><font color="Maroon" title="c2">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="finseq_3.html#K1" title="FINSEQ_3:func.1">-</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font></span><br/>



<b>set </b><font color="Maroon" title="c3">q</font> =  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:65"/><i><font color="Green" title="E45">A1</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>thus </b><a NAME="E2:65"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="finseq_3.html#K1" title="FINSEQ_3:func.1">-</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font>
  <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="finseq_3.html#K1" title="FINSEQ_3:func.1">-</a> <font color="Maroon" title="c2">A</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="finseq_3.html#K1" title="FINSEQ_3:func.1">-</a> <font color="Maroon" title="c2">A</font></span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<b>assume </b><a NAME="E1:65_1"/><i><font color="Green" title="E46">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="finseq_3.html#K1" title="FINSEQ_3:func.1">-</a> <font color="Maroon" title="c2">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font></span><br/>

<a NAME="E2:65_1"/><b>then </b><i><font color="Green" title="E47">A3</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:65_1_1"/><i><font color="Green" title="E48">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>consider </b><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:65_1_1"/><i><font color="Green" title="E49">A5</font></i>: 
<font color="Maroon" title="c5">y</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"><font color="Maroon" title="c1">p</font> <a href="finseq_3.html#K1" title="FINSEQ_3:func.1">-</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <b>and </b><br/><a NAME="E4:65_1_1"/><i><font color="Green" title="E50">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="finseq_3.html#K1" title="FINSEQ_3:func.1">-</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E1:65_1"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>set </b><font color="Maroon" title="c6">z</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font>;<br/><a NAME="E5:65_1_1"/><i><font color="Green" title="E51">A7</font></i>: 
<font color="Maroon" title="c5">y</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="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_3.html#E1:65"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="finseq_3.html#E3:65_1_1"><i><font color="Green" title="E49">A5</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E6:65_1_1"/><b>then </b><i><font color="Green" title="E52">A8</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:65_1_1"/><i><font color="Green" title="E53">A9</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</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="c1">p</font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E1:65"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="finseq_3.html#E3:65_1_1"><i><font color="Green" title="E49">A5</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E8:65_1_1"/><i><font color="Green" title="E54">A10</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="finseq_3.html#K1" title="FINSEQ_3:func.1">-</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_3.html#E1:65"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="finseq_3.html#E5:65_1_1"><i><font color="Green" title="E51">A7</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E9:65_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><a NAME="E10:65_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_3.html#E6:65_1_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E11:65_1_1"/><b>then </b>
not <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">p</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E12:65_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="finseq_3.html#E1:65_1_1"><i><font color="Green" title="E48">A4</font></i></a>, <a class="txt" href="finseq_3.html#E4:65_1_1"><i><font color="Green" title="E50">A6</font></i></a>, <a class="txt" href="finseq_3.html#E7:65_1_1"><i><font color="Green" title="E53">A9</font></i></a>, <a class="txt" href="finseq_3.html#E8:65_1_1"><i><font color="Green" title="E54">A10</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:65_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"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E2:65_1"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="finseq_3.html#K1" title="FINSEQ_3:func.1">-</a> <font color="Maroon" title="c2">A</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E3:65"/><i><font color="Green" title="E46">A11</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="finseq_3.html#K1" title="FINSEQ_3:func.1">-</a> <font color="Maroon" title="c2">A</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:65"/><i><font color="Green" title="E47">A12</font></i>: 
<font color="Maroon" title="c5">y</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="c1">p</font>
 <b>and </b><br/><a NAME="E6:65"/><i><font color="Green" title="E48">A13</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:65"/>
not <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="finseq_3.html#E3:65"><i><font color="Green" title="E46">A11</font></i></a>, <a class="txt" href="finseq_3.html#E6:65"><i><font color="Green" title="E48">A13</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E8:65"/><b>then </b><i><font color="Green" title="E49">A14</font></i>: 
not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">p</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E9:65"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E10:65"/><b>then </b>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">A</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/>
<a NAME="E11:65"/><b>then </b>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_3.html#E5:65"><i><font color="Green" title="E47">A12</font></i></a>, <a class="txt" href="finseq_3.html#E8:65"><i><font color="Green" title="E49">A14</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:65"/><i><font color="Green" title="E50">A15</font></i>: 
<font color="Maroon" title="c6">z</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="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E14:65"/><i><font color="Green" title="E51">A16</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E15:65"/><i><font color="Green" title="E52">A17</font></i>: 
<font color="Maroon" title="c6">z</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"><font color="Maroon" title="c1">p</font> <a href="finseq_3.html#K1" title="FINSEQ_3:func.1">-</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_3.html#E1:65"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="finseq_3.html#E5:65"><i><font color="Green" title="E47">A12</font></i></a>, <a class="txt" href="finseq_3.html#E13:65"><i><font color="Green" title="E50">A15</font></i></a>, <a class="txt" href="finseq_3.html#E14:65"><i><font color="Green" title="E51">A16</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<a NAME="E16:65"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="finseq_3.html#K1" title="FINSEQ_3:func.1">-</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="finseq_3.html#E1:65"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="finseq_3.html#E6:65"><i><font color="Green" title="E48">A13</font></i></a>, <a class="txt" href="finseq_3.html#E13:65"><i><font color="Green" title="E50">A15</font></i></a>, <a class="txt" href="finseq_3.html#E14:65"><i><font color="Green" title="E51">A16</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<b>hence </b><a NAME="E17:65"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="finseq_3.html#K1" title="FINSEQ_3:func.1">-</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_3.html#E15:65"><i><font color="Green" title="E52">A17</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
