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<span class="kw">let </span><font color="Maroon" title="c1">Y</font>, <font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <font color="Maroon" title="c2">X</font> implies  ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( not <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="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="Olive" title="b1">A</font></span>)</span> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:28"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c1">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E2:28"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( not <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="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="Olive" title="b1">A</font></span>)</span> )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <span class="kw">means </span><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>,$1;<br/>
<a NAME="E3:28"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
 ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st  <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="classes1.html#T22" target="_self" title="CLASSES1:th.22">Th22</a></span>;<br/>
<a NAME="E4:28"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
 ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E21">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E6:28"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">A</font>] &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] holds <br/><font color="Maroon" title="c3">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">B</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal1.html#S1" title="ORDINAL1:sch.1">ORDINAL1:sch 1</a>(<a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E22">A4</font></span></a>);<br/></span>
<a NAME="E7:28"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
not <font color="Maroon" title="c1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E8:28"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
 <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c2">X</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="classes1.html#E15"><span class="lab"><font color="Green" title="E10">Lm1</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E26">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:28_1"/><span class="lab"><font color="Green" title="E27">A9</font></span>: 
<font color="Maroon" title="c3">A</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:28_1"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
 ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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="Olive" title="b1">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E7:28"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="ref" href="classes1.html#T16" target="_self" title="CLASSES1:th.16">Th16</a></span>;<br/><span class="kw">thus </span><a NAME="E3:28_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E2:28_1"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c4">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E11:28"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:28"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">B</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <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">B</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>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">B</font></span>)</span> )</span><br/>

<a NAME="E12:28"/><span class="lab"><font color="Green" title="E28">A12</font></span>: 
not <font color="Maroon" title="c3">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:28"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<span class="kw">thus </span><a NAME="E13:28"/>
( not <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">B</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>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">B</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E11:28"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E12:28"><span class="lab"><font color="Green" title="E28">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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