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<b>let </b><font color="Maroon">B</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:47"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<div><i><font color="Green">E21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><a NAME="E1:47_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font>
 ;<br/><b>thus </b> <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font></span>)</span> = 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">C</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T30" target="_self">Th30</a>, <a class="ref" href="ordinal2.html#T45">ORDINAL2:45</a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T2">ORDINAL2:2</a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T30" target="_self">Th30</a>, <a class="ref" href="ordinal2.html#T2">ORDINAL2:2</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_2"/>
for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  holds not <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">C</font>
 ;<br/><a NAME="E2:47_2"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">B</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/><a NAME="E3:47_2"/><b>then </b>
<font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 <b>by </b><i><a class="ref" href="ordinal3.html#T27" target="_self">Th27</a>, </i>;<br/><a NAME="E4:47_2"/><b>then </b>
(  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font> &amp;  <a href="tarski.html#K3">union</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="ordinal3.html#T30" target="_self">Th30</a>, <a class="ref" href="ordinal1.html#D6">ORDINAL1:def 6</a></i>;<br/><b>hence </b><a NAME="E5:47_2"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">B</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:47"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="ordinal3.html#T29" target="_self">Th29</a></i>;<br/>


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