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<b>let </b><font color="Maroon" title="c1">Y</font>, <font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Z</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"> for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">A</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>  &amp; <font color="Olive" title="b1">A</font> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> &amp; <font color="Maroon" title="c1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c2">X</font>,<font color="Olive" title="b1">A</font> &amp; ( <font color="Maroon" title="c3">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Y</font> or <font color="Maroon" title="c3">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Y</font> ) holds <br/><font color="Maroon" title="c3">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c2">X</font>,<font color="Olive" title="b1">A</font></span><br/><b>let </b><font color="Maroon" title="c4">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"> ( <font color="Maroon" title="c4">A</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>  &amp; <font color="Maroon" title="c4">A</font> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> &amp; <font color="Maroon" title="c1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c4">A</font> &amp; ( <font color="Maroon" title="c3">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Y</font> or <font color="Maroon" title="c3">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Y</font> ) implies <font color="Maroon" title="c3">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c4">A</font> )</span><br/>







<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c4">A</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>  &amp; <font color="Maroon" title="c4">A</font> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> &amp; <font color="Maroon" title="c1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c4">A</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="c3">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Y</font> &amp; not <font color="Maroon" title="c3">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Y</font> ) or <font color="Maroon" title="c3">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c4">A</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c5">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:18"/><i><font color="Green" title="E15">A2</font></i>: 
( <font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c5">B</font> )
 <b>by </b><i><a class="ref" href="classes1.html#T16" target="_self" title="CLASSES1:th.16">Th16</a></i>;<br/>
<a NAME="E4:18"/><i><font color="Green" title="E16">A3</font></i>: 
 <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c2">X</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:18"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="classes1.html#T13" target="_self" title="CLASSES1:th.13">Th13</a></i>;<br/>
<a NAME="E5:18"/><i><font color="Green" title="E17">A4</font></i>: 
( <font color="Maroon" title="c3">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Y</font> implies <font color="Maroon" title="c3">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c2">X</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">B</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E3:18"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="classes1.html#T13" target="_self" title="CLASSES1:th.13">Th13</a></i>;<br/>
<a NAME="E6:18"/><i><font color="Green" title="E18">A5</font></i>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 
<b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E3:18"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></i>;<br/>
<b>assume </b><a NAME="E7:18"/>
( <font color="Maroon" title="c3">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Y</font> or <font color="Maroon" title="c3">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c4">A</font></span><br/>

<b>hence </b><a NAME="E8:18"/>
<font color="Maroon" title="c3">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E4:18"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E5:18"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="#E6:18"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="classes1.html#T16" target="_self" title="CLASSES1:th.16">Th16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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