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<b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">C1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">D1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">C2</font> be   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">D2</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:50"/><i><font color="Green">E19</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">D1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">D2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<b>set </b><font color="Maroon">B</font> = <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D1</font>;<br/>
<div><i><font color="Green">E21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1"/>
<font color="Maroon">C1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font>
 ;<br/><b>then consider </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:50_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D2</font>
<b>by </b><i>, , <a class="ref" href="ordinal3.html#T27" target="_self">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D2</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T1" target="_self">Th1</a></i>
;<br/><a NAME="E5:50_1"/><b>then </b>
( <font color="Maroon">D1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font> &amp; <font color="Maroon">C</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D2</font> )
 <b>by </b><i>, , , <a class="txt" href="ordinal3.html#E2:50"><i><font color="Green">E21</font></i></a></i>;<br/><b>hence </b><a NAME="E6:50_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_2"/>
<font color="Maroon">C2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font>
 ;<br/><b>then consider </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:50_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C2</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D1</font>
<b>by </b><i>, <a class="ref" href="ordinal3.html#T27" target="_self">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D1</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T1" target="_self">Th1</a></i>
;<br/><a NAME="E5:50_2"/><b>then </b>
( <font color="Maroon">D2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font> &amp; <font color="Maroon">C</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D1</font> )
 <b>by </b><i><a class="txt" href="ordinal3.html#E3:50_2"><i><font color="Green">E35</font></i></a>, , , , <a class="txt" href="ordinal3.html#E2:50"><i><font color="Green">E21</font></i></a></i>;<br/><b>hence </b><a NAME="E6:50_2"/>
contradiction
 <b>by </b><i><a class="txt" href="ordinal3.html#E3:50_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:50"/>
<font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T24">ORDINAL1:24</a></i>;<br/>

<b>hence </b><a NAME="E5:50"/>
<font color="Maroon">D1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font>
 <b>by </b><i>, </i>;<br/>


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