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<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c3">D1</font>, <font color="Maroon" title="c4">C2</font>, <font color="Maroon" title="c5">D2</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:49"/><i><font color="Green" title="E37">A1</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">C1</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C2</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c5">D2</font> &amp; <font color="Maroon" title="c3">D1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">D2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c6">B</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">C1</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">D1</font>;<br/>
<div><i><font color="Green" title="E38">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1"/>
<font color="Maroon" title="c2">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C2</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:49_1"/><i><font color="Green" title="E39">A3</font></i>: 
( <font color="Maroon" title="c4">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c7">C</font> &amp; <font color="Maroon" title="c7">C</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="ordinal3.html#T31" target="_self" title="ORDINAL3:th.31">Th31</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">C1</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">D1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">C1</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c5">D2</font>
<b>by </b><i><a class="txt" href="ordinal3.html#E1:49"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="ordinal3.html#E3:49_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="ordinal3.html#T54" target="_self" title="ORDINAL3:th.54">Th54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">C1</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c5">D2</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T33" target="_self" title="ORDINAL3:th.33">Th33</a></i>
;<br/><a NAME="E5:49_1"/><b>then </b>
( <font color="Maroon" title="c3">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c5">D2</font> &amp; <font color="Maroon" title="c1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c5">D2</font> )
 <b>by </b><i><a class="txt" href="ordinal3.html#E3:49_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="ordinal3.html#T24" target="_self" title="ORDINAL3:th.24">Th24</a>, <a class="ref" href="ordinal3.html#T27" target="_self" title="ORDINAL3:th.27">Th27</a>, <a class="ref" href="ordinal3.html#T39" target="_self" title="ORDINAL3:th.39">Th39</a></i>;<br/><b>hence </b><a NAME="E6:49_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ordinal3.html#E1:49"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_2"/>
<font color="Maroon" title="c4">C2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C1</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:49_2"/><i><font color="Green" title="E39">A4</font></i>: 
( <font color="Maroon" title="c2">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C2</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c7">C</font> &amp; <font color="Maroon" title="c7">C</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="ordinal3.html#T31" target="_self" title="ORDINAL3:th.31">Th31</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">C1</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">D1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">C2</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">D1</font>
<b>by </b><i><a class="txt" href="ordinal3.html#E3:49_2"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="ordinal3.html#T54" target="_self" title="ORDINAL3:th.54">Th54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">C2</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">D1</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T33" target="_self" title="ORDINAL3:th.33">Th33</a></i>
;<br/><a NAME="E5:49_2"/><b>then </b>
( <font color="Maroon" title="c5">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">D1</font> &amp; <font color="Maroon" title="c1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">D1</font> )
 <b>by </b><i><a class="txt" href="ordinal3.html#E1:49"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="ordinal3.html#E3:49_2"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="ordinal3.html#T24" target="_self" title="ORDINAL3:th.24">Th24</a>, <a class="ref" href="ordinal3.html#T27" target="_self" title="ORDINAL3:th.27">Th27</a>, <a class="ref" href="ordinal3.html#T39" target="_self" title="ORDINAL3:th.39">Th39</a></i>;<br/><b>hence </b><a NAME="E6:49_2"/>
contradiction
 <b>by </b><i><a class="txt" href="ordinal3.html#E1:49"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:49"/>
<font color="Maroon" title="c2">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C2</font>
 <b>by </b><i><a class="txt" href="ordinal3.html#E2:49"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="ordinal1.html#T24" title="ORDINAL1:th.24">ORDINAL1:24</a></i>;<br/>

<b>hence </b><a NAME="E5:49"/>
<font color="Maroon" title="c3">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font>
 <b>by </b><i><a class="txt" href="ordinal3.html#E1:49"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="ordinal3.html#T24" target="_self" title="ORDINAL3:th.24">Th24</a></i>;<br/>


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