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<span class="kw">consider </span><font color="Maroon" title="c1">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E2:48_1"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">A</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E21">A1</font></span></a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] );<br/>
<span class="kw">consider </span><font color="Maroon" title="c2">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:48_1"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></span>
<a NAME="E5:48_1"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a></span>;<br/>
<a NAME="E6:48_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E7:48_1"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T10" target="_self" title="ORDINAL1:th.10">Th10</a></span>;<br/>
<a NAME="E8:48_1"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E4:48_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E10:48_1"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
<font color="Maroon" title="c3">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E11:48_1"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/><font color="Maroon" title="c3">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="ordinal1.html#T32" target="_self" title="ORDINAL1:th.32">Th32</a></span>;<br/>
<a NAME="E12:48_1"/>
<font color="Maroon" title="c3">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E10:48_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a></span>;<br/>
<a NAME="E13:48_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A7</font></span>: 
<font color="Maroon" title="c3">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" target="_self" title="ORDINAL1:def.2">Def2</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">C</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">P<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] holds <br/><font color="Maroon" title="c3">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">B</font> ) )</span><br/>

<a NAME="E14:48_1"/>
 ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E10:48_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:48_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">C</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] holds <br/><font color="Maroon" title="c3">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">B</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</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">P<sub>1</sub></font>[<font color="Maroon" title="c4">B</font>] implies <font color="Maroon" title="c3">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E16:48_1"/><span class="lab"><font color="Green" title="E28">A8</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<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"> <font color="Maroon" title="c3">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">B</font></span><br/>

<span class="kw">assume </span><a NAME="E17:48_1"/><span class="lab"><font color="Green" title="E29">A9</font></span>: 
not <font color="Maroon" title="c3">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <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"> contradiction</span><br/>

<a NAME="E18:48_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">B</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c3">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<a NAME="E19:48_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T21" target="_self" title="ORDINAL1:th.21">Th21</a></span>;<br/>
<a NAME="E20:48_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E16:48_1"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="txt" href="#E13:48_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:48_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E17:48_1"><span class="lab"><font color="Green" title="E29">A9</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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