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

<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</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">X</font>, <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y</font> holds <br/> <a href="ordinal2.html#K2" title="ORDINAL2:func.2">inf</a> <font color="Olive" title="b2">Y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K2" title="ORDINAL2:func.2">inf</a> <font color="Olive" title="b1">X</font></span><br/><b>let </b><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</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"> ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> implies  <a href="ordinal2.html#K2" title="ORDINAL2:func.2">inf</a> <font color="Maroon" title="c3">Y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K2" title="ORDINAL2:func.2">inf</a> <font color="Maroon" title="c2">X</font> )</span><br/>





<b>assume </b><a NAME="E1:19"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> or  <a href="ordinal2.html#K2" title="ORDINAL2:func.2">inf</a> <font color="Maroon" title="c3">Y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K2" title="ORDINAL2:func.2">inf</a> <font color="Maroon" title="c2">X</font> )</span><br/>

<a NAME="E2:19"/><b>then </b><i><font color="Green" title="E8">A1</font></i>: 
 <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a></i>;<br/>
<b>assume </b><a NAME="E3:19"/>
<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="ordinal2.html#K2" title="ORDINAL2:func.2">inf</a> <font color="Maroon" title="c3">Y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K2" title="ORDINAL2:func.2">inf</a> <font color="Maroon" title="c2">X</font></span><br/>

<a NAME="E4:19"/><b>then </b>
(  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c3">Y</font> &amp;  <a href="ordinal2.html#K2" title="ORDINAL2:func.2">inf</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp;  <a href="ordinal2.html#K2" title="ORDINAL2:func.2">inf</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c3">Y</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T11" target="_self" title="ORDINAL2:th.11">Th11</a></i>;<br/>
<b>hence </b><a NAME="E5:19"/>
 <a href="ordinal2.html#K2" title="ORDINAL2:func.2">inf</a> <font color="Maroon" title="c3">Y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K2" title="ORDINAL2:func.2">inf</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="#E2:19"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="setfam_1.html#T7" title="SETFAM_1:th.7">SETFAM_1:7</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>
