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

<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="exchsort.html#NM1" title="EXCHSORT:NM.1">array</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="exchsort.html#K2" title="EXCHSORT:func.2">limit-</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="exchsort.html#K1" title="EXCHSORT:func.1">base-</a> <font color="Maroon" title="c1">A</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> being   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E2:33"/><span class="lab"><font color="Green" title="E18">A0</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="exchsort.html#D1" target="_self" title="EXCHSORT:def.1">SEG</a></span>;<br/>
<a NAME="E3:33"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
 <a href="exchsort.html#K2" title="EXCHSORT:func.2">limit-</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="exchsort.html#T16" target="_self" title="EXCHSORT:th.16">Th005</a></span>;<br/>
<a NAME="E4:33"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
 <a href="exchsort.html#K1" title="EXCHSORT:func.1">base-</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K2" title="ORDINAL2:func.2">inf</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="exchsort.html#T20" target="_self" title="EXCHSORT:th.20">Th007</a></span>;<br/>
<span class="kw">thus </span><a NAME="E5:33"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="exchsort.html#K2" title="EXCHSORT:func.2">limit-</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="exchsort.html#K1" title="EXCHSORT:func.1">base-</a> <font color="Maroon" title="c1">A</font></span>)</span>
  <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="exchsort.html#K2" title="EXCHSORT:func.2">limit-</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="exchsort.html#K1" title="EXCHSORT:func.1">base-</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font> 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="exchsort.html#K2" title="EXCHSORT:func.2">limit-</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="exchsort.html#K1" title="EXCHSORT:func.1">base-</a> <font color="Maroon" title="c1">A</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="exchsort.html#K2" title="EXCHSORT:func.2">limit-</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="exchsort.html#K1" title="EXCHSORT:func.1">base-</a> <font color="Maroon" title="c1">A</font></span>)</span></span><br/>

<a NAME="E2:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A3</font></span>: 
(  <a href="exchsort.html#K1" title="EXCHSORT:func.1">base-</a> <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="exchsort.html#K2" title="EXCHSORT:func.2">limit-</a> <font color="Maroon" title="c1">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="ordinal2.html#T14" title="ORDINAL2:th.14">ORDINAL2:14</a>, <a class="ref" href="ordinal2.html#T19" title="ORDINAL2:th.19">ORDINAL2:19</a></span>;<br/>
<a NAME="E3:33_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="exchsort.html#K1" title="EXCHSORT:func.1">base-</a> <font color="Maroon" title="c1">A</font> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">x</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E4:33_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="exchsort.html#K2" title="EXCHSORT:func.2">limit-</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="exchsort.html#K1" title="EXCHSORT:func.1">base-</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></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="exchsort.html#K2" title="EXCHSORT:func.2">limit-</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="exchsort.html#K1" title="EXCHSORT:func.1">base-</a> <font color="Maroon" title="c1">A</font></span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E6:33"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
<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="exchsort.html#K2" title="EXCHSORT:func.2">limit-</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="exchsort.html#K1" title="EXCHSORT:func.1">base-</a> <font color="Maroon" title="c1">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<a NAME="E8:33"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">x</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="ordinal2.html#T21" title="ORDINAL2:th.21">ORDINAL2:21</a></span>;<br/>
<a NAME="E9:33"/><span class="kw">then </span>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="exchsort.html#K2" title="EXCHSORT:func.2">limit-</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="exchsort.html#K1" title="EXCHSORT:func.1">base-</a> <font color="Maroon" title="c1">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E18">A0</font></span></a>, <a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="exchsort.html#T6" target="_self" title="EXCHSORT:th.6">Th008</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:33"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E18">A0</font></span></a>, <a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E21">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
